A soft catching glove does not make an incoming ball’s momentum disappear. It spreads the required change over more time. Collisions become easier to understand when we first account for momentum, then ask separately where the energy went.
We will build impulse from a force–time graph, distinguish elastic and sticking collisions, derive centre-of-mass motion and extend the method to two dimensions. Energy checks will show what momentum conservation does—and does not—determine.
What you will learn to do
- Add momentum as a vector and interpret impulse as its change.
- Assess external impulse before applying conservation.
- Solve sticking and elastic one-dimensional impacts.
- Use mass-weighted centre-of-mass position and velocity.
- Resolve two-dimensional collisions and identify missing information.
- Separate collision dynamics from later energy-conserving motion.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions Learning goals: - Add momentum as a vector and interpret impulse as its change. - Assess external impulse before applying conservation. - Solve sticking and elastic one-dimensional impacts. - Use mass-weighted centre-of-mass position and velocity. - Resolve two-dimensional collisions and identify missing information. - Separate collision dynamics from later energy-conserving motion. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
For the same change in momentum, what happens to average force when stopping time increases?
Compare your reasoning
Its required magnitude decreases because impulse is average force multiplied by duration. This does not mean every detail of the force–time curve is fixed.
Revisit the explanation →01 / Impulse, momentum and collisions
Momentum keeps the direction that kinetic energy loses
A particle’s momentum is p = mv. Because velocity is a vector, momentum is a vector too. Two equal masses moving equally fast in opposite directions have momenta that cancel, while their positive kinetic energies add. Zero total momentum therefore does not imply that the particles are stationary or that the system has zero kinetic energy.
In classical mechanics with constant mass, Newton’s second law can be written Fnet = dp/dt. A force changes momentum through its duration of action. This makes momentum useful for short interactions whose detailed forces are complicated, such as impacts.
For a system of particles, add the momentum vectors in a common frame. Internal forces exchange momentum between parts. Under the usual classical particle-interaction assumptions, equal and opposite internal forces cancel in the total rate equation, leaving the net external force to change the total momentum.
The system boundary must be specified. A force internal to a two-body system is external when only one body is selected.
WORKED EXAMPLE
A moving system with zero total momentum
A 2 kg cart moves right at 3 m/s while a 3 kg cart moves left at 2 m/s. Find the total momentum and total kinetic energy in the ground frame.
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STOP & THINK
A system has zero total momentum in a chosen frame. What must be true?
- Every particle is at rest
- The mass-weighted velocity sum is zero
- Its total kinetic energy is zero
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions
Focus: Momentum keeps the direction that kinetic energy loses
Learning goals:
- Add momentum as a vector and interpret impulse as its change.
- Assess external impulse before applying conservation.
- Solve sticking and elastic one-dimensional impacts.
- Use mass-weighted centre-of-mass position and velocity.
- Resolve two-dimensional collisions and identify missing information.
- Separate collision dynamics from later energy-conserving motion.
Equations in this section (LaTeX):
\mathbf P=\sum_i m_i\mathbf v_i,\qquad\frac{d\mathbf P}{dt}=\mathbf F_{\mathrm{ext,net}}
Meaning: The system boundary must be specified. A force internal to a two-body system is external when only one body is selected.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Impulse, momentum and collisions
A force–time area gives a momentum change
Integrating force over time gives impulse J = ∫F dt. For the net force on a particle, impulse equals its momentum change. Each component can be integrated independently, preserving signs. The units N·s are identical to kg·m/s.
A force–time graph supplies an impulse through its signed area. A triangular pulse has half the area of a rectangle with the same peak and duration. Average force is the impulse divided by the duration; it need not equal the peak force. A longer interaction can produce the same impulse with a smaller average force.
Distinguish the impulse of one contact from the net impulse. During a vertical bounce, both the surface contact and gravity act. If the contact is very brief, gravity’s impulse may be negligible compared with the contact impulse, but this is an approximation to justify, not a change in the force law.
Work integrates over displacement and changes kinetic energy; impulse integrates over time and changes momentum.
Explain this equation
Read it in words
Net impulse, the accumulated net force over time, equals the change in momentum. Average net force is that impulse divided by the contact time.
Symbols and units
- Net impulse vector; this J is a quantity symbol, not the joule unit.N·s, equivalently kg·m/s.
- Vector sum of forces acting on the object at each instant.Newtons (N).
- Initial time, final time and positive elapsed time t_f − t_i.Seconds (s).
- Initial and final momentum vectors. For a constant-mass particle p = mv.kg·m/s.
- Constant force that would produce the same impulse over the same interval.Newtons (N).
Signs and operations
The integral is the signed area under each force–time component graph. The subtraction is final momentum minus initial momentum; reversing velocity is not the same as merely reducing speed.
When it applies
Use net force on the same object throughout. For a system, use external force for its total momentum change. Average force need not equal the peak force.
Make it concrete
A 0.20 kg ball changes velocity from +5 to −3 m/s. Its impulse is −1.6 N·s. Over 0.020 s, the average net force is −80 N; the minus sign gives its direction.
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Equation (LaTeX): \mathbf J_{\mathrm{net}}=\int_{t_i}^{t_f}\mathbf F_{\mathrm{net}}\,dt=\mathbf p_f-\mathbf p_i,\qquad\mathbf F_{\mathrm{avg}}=\frac{\mathbf J}{\Delta t}
Read it as: Net impulse, the accumulated net force over time, equals the change in momentum. Average net force is that impulse divided by the contact time.
Symbols:
\mathbf J_{\mathrm{net}}: Net impulse vector; this J is a quantity symbol, not the joule unit.; N·s, equivalently kg·m/s.
\mathbf F_{\mathrm{net}}: Vector sum of forces acting on the object at each instant.; Newtons (N).
t_i,\ t_f,\ \Delta t: Initial time, final time and positive elapsed time t_f − t_i.; Seconds (s).
\mathbf p_i,\ \mathbf p_f: Initial and final momentum vectors. For a constant-mass particle p = mv.; kg·m/s.
\mathbf F_{\mathrm{avg}}: Constant force that would produce the same impulse over the same interval.; Newtons (N).
Signs and operations: The integral is the signed area under each force–time component graph. The subtraction is final momentum minus initial momentum; reversing velocity is not the same as merely reducing speed.
Conditions: Use net force on the same object throughout. For a system, use external force for its total momentum change. Average force need not equal the peak force.
Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.WORKED EXAMPLE
Use the area, not the peak alone
A 2 kg cart initially moves at 1 m/s along +x. Its net horizontal force is a triangular pulse lasting 0.40 s, starting and ending at zero and reaching a peak of 30 N. Find the impulse, average force and final velocity.
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Compare two pulses with the same impulse
- Short pulse
- 0
- Long pulse
- 0
STOP & THINK
A 0.20 kg ball changes velocity from +10 m/s to −5 m/s along one line. What is its signed net impulse?
Answer in N·s.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions
Focus: A force–time area gives a momentum change
Learning goals:
- Add momentum as a vector and interpret impulse as its change.
- Assess external impulse before applying conservation.
- Solve sticking and elastic one-dimensional impacts.
- Use mass-weighted centre-of-mass position and velocity.
- Resolve two-dimensional collisions and identify missing information.
- Separate collision dynamics from later energy-conserving motion.
Equations in this section (LaTeX):
\mathbf J_{\mathrm{net}}=\int_{t_i}^{t_f}\mathbf F_{\mathrm{net}}\,dt=\mathbf p_f-\mathbf p_i,\qquad\mathbf F_{\mathrm{avg}}=\frac{\mathbf J}{\Delta t}
Meaning: Work integrates over displacement and changes kinetic energy; impulse integrates over time and changes momentum.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.03 / Impulse, momentum and collisions
Decide whether external impulse is negligible
Total momentum is conserved over an interval if the net external impulse over that interval is zero. It may be approximately conserved when that impulse is small enough for the required accuracy. Saying “there is a collision” does not automatically establish this condition.
An impact between two carts on a nearly smooth horizontal track can have a large mutual contact impulse while the external horizontal impulse is small. Horizontal momentum can then be conserved. The vertical normal and weight may each be large, but their effects do not supply a significant horizontal impulse. Conservation can therefore hold component by component even when another component requires a separate balance.
A wall collision illustrates the importance of the system. A ball alone receives an external impulse from the wall, so its momentum is not conserved. Including the wall and Earth makes the contact internal, but the vastly larger mass has a tiny recoil velocity. Treating the wall as fixed is a useful approximation; it is not evidence that the ball conserves momentum by itself.
WORKED EXAMPLE
Quantify an omitted impulse before neglecting it
Two colliding carts exchange a horizontal contact impulse of about 4 N·s over 0.020 s. A constant external horizontal resistance of 0.50 N acts on the two-cart system. Estimate the change of total horizontal momentum due to that resistance and compare it with the contact-impulse scale.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions Focus: Decide whether external impulse is negligible Learning goals: - Add momentum as a vector and interpret impulse as its change. - Assess external impulse before applying conservation. - Solve sticking and elastic one-dimensional impacts. - Use mass-weighted centre-of-mass position and velocity. - Resolve two-dimensional collisions and identify missing information. - Separate collision dynamics from later energy-conserving motion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Impulse, momentum and collisions
Momentum survives a sticking collision when kinetic energy does not
In a perfectly inelastic collision, the colliding bodies have a common velocity immediately afterwards. With negligible external impulse, momentum conservation determines that velocity. Kinetic energy usually decreases because relative-motion energy is converted into deformation, heat, sound or other internal forms.
Do not impose both kinetic-energy conservation and sticking unless a special initial state makes them compatible. For two particles already sharing a velocity, there is no relative motion to remove; otherwise an ordinary sticking event loses translational kinetic energy. Total energy accounting can still be satisfied when internal energy is included.
The common velocity is a mass-weighted average of the initial velocities. It must lie between them in one dimension for positive masses. That bound is an immediate check on a result, including when the initial directions are opposite.
WORKED EXAMPLE
Find the common velocity and account for the lost kinetic energy
A 2 kg cart moving at +6 m/s sticks to a 4 kg cart initially at rest on a smooth horizontal track. Find their common velocity and the loss of translational kinetic energy.
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STOP & THINK
Why is kinetic-energy conservation generally inappropriate for a sticking collision?
- Momentum conservation stops applying
- Relative translational motion can become internal energy
- Energy can simply disappear
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions Focus: Momentum survives a sticking collision when kinetic energy does not Learning goals: - Add momentum as a vector and interpret impulse as its change. - Assess external impulse before applying conservation. - Solve sticking and elastic one-dimensional impacts. - Use mass-weighted centre-of-mass position and velocity. - Resolve two-dimensional collisions and identify missing information. - Separate collision dynamics from later energy-conserving motion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Impulse, momentum and collisions
A collision model supplies the missing relation
Momentum conservation alone usually does not determine two separate final velocities. An elastic collision supplies the additional condition that total translational kinetic energy is conserved. In one dimension, combining the two laws shows that relative separation speed equals relative approach speed.
More generally, the coefficient of restitution e is the relative separation speed divided by relative approach speed along the line of impact. For a standard passive one-dimensional impact, 0 ≤ e ≤ 1; e = 1 is elastic and e = 0 removes the normal relative velocity. Energy-releasing impacts can require a different model and need not obey that passive bound.
Use signed velocities consistently. If body 1 approaches body 2 with u₁ > u₂ along +x, then v₂ − v₁ = e(u₁ − u₂). In an oblique smooth-sphere collision this relation applies along the collision normal, while tangential components require their own impulse conditions. Do not apply it to full speed magnitudes in arbitrary directions.
WORKED EXAMPLE
Two equations determine two final velocities
A 1 kg cart moving at +6 m/s collides elastically in one dimension with a 2 kg cart initially at rest. External impulse is negligible. Find both final velocities.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions Focus: A collision model supplies the missing relation Learning goals: - Add momentum as a vector and interpret impulse as its change. - Assess external impulse before applying conservation. - Solve sticking and elastic one-dimensional impacts. - Use mass-weighted centre-of-mass position and velocity. - Resolve two-dimensional collisions and identify missing information. - Separate collision dynamics from later energy-conserving motion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Impulse, momentum and collisions
The centre of mass is a mass-weighted position
The centre of mass is the position obtained by weighting each particle’s position by its mass and dividing by the total mass. For positive masses it lies within the span of their positions along any chosen axis, closer to the heavier concentration. It need not coincide with any material particle; the centre of a uniform ring is in its empty interior.
For a fixed collection of particles with constant total mass, differentiate the position definition to obtain Vcom = P/M. A second derivative gives M Acom = Fext,net. The centre of mass therefore responds to the external resultant even while the parts collide, separate or change their relative arrangement.
Internal forces can strongly change individual motions without changing the total momentum. If the external resultant is zero, the centre of mass moves with constant velocity. It remains fixed only in a frame where that velocity is initially zero. Do not replace “constant velocity” with “at rest” unless the initial condition supports it.
These relations use the same set of particles throughout. An open system exchanging mass requires additional momentum-flow accounting.
WORKED EXAMPLE
Position and velocity have separate weighted averages
At an instant, a 2 kg particle is at x = 0 m moving at +4 m/s and a 6 kg particle is at x = 4 m moving at −2 m/s. Find the centre-of-mass position and velocity.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions
Focus: The centre of mass is a mass-weighted position
Learning goals:
- Add momentum as a vector and interpret impulse as its change.
- Assess external impulse before applying conservation.
- Solve sticking and elastic one-dimensional impacts.
- Use mass-weighted centre-of-mass position and velocity.
- Resolve two-dimensional collisions and identify missing information.
- Separate collision dynamics from later energy-conserving motion.
Equations in this section (LaTeX):
\mathbf R=\frac{\sum_i m_i\mathbf r_i}{M},\qquad\mathbf V_{\mathrm{com}}=\frac{\mathbf P}{M},\qquad M\mathbf A_{\mathrm{com}}=\mathbf F_{\mathrm{ext,net}}
Meaning: These relations use the same set of particles throughout. An open system exchanging mass requires additional momentum-flow accounting.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.07 / Impulse, momentum and collisions
Internal motion changes positions around the moving centre
A person walking along a freely floating boat pushes the boat backward while moving forward relative to it. If external horizontal impulse is negligible, the combined horizontal momentum stays unchanged. If both initially rest in the ground frame, their centre of mass stays at its initial horizontal position.
The distance walked relative to the boat is not the person’s ground displacement. Let the boat move by Δxb and the person walk a distance d relative to it. Then the person’s ground displacement is Δxp = Δxb + d. Combine this geometric relation with the unchanged centre of mass.
The same principle helps analyse recoil and explosions. Internal energy can increase the kinetic energy of separating parts without changing the total momentum. If uniform gravity is the only external force, the centre of mass follows the corresponding gravitational trajectory even though individual fragments separate. Ground contact or unequal drag later changes that external-force model.
WORKED EXAMPLE
Separate walking distance from ground displacement
A 40 kg person walks 3 m to the right relative to a 60 kg boat. Initially both are at rest. Neglect external horizontal forces and treat the boat as translating without rotation. Find the boat’s displacement relative to shore.
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STOP & THINK
Two particles of masses 1 kg and 3 kg lie at x = 0 m and x = 8 m respectively. What is their centre-of-mass coordinate?
Answer in m.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions Focus: Internal motion changes positions around the moving centre Learning goals: - Add momentum as a vector and interpret impulse as its change. - Assess external impulse before applying conservation. - Solve sticking and elastic one-dimensional impacts. - Use mass-weighted centre-of-mass position and velocity. - Resolve two-dimensional collisions and identify missing information. - Separate collision dynamics from later energy-conserving motion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
08 / Impulse, momentum and collisions
Conserve each momentum component in the same frame
In two dimensions, total momentum conservation supplies two scalar equations: one for x and one for y. Kinetic energy, if conserved, supplies one additional scalar equation. Do not add speed magnitudes as though both outgoing bodies must travel in the initial direction.
Count unknowns before solving. Two unknown final velocity vectors contain four scalar components. Two momentum equations and one elastic-energy equation may still leave one degree of freedom unless the collision geometry, a scattering angle or another condition is specified. A drawing alone should not silently supply missing data.
For a sticking collision, the final common vector has only two components, so the two momentum equations can determine it directly. Afterwards calculate its magnitude and direction. Use the component signs to select the correct quadrant rather than trusting an inverse tangent that returns an ambiguous angle.
WORKED EXAMPLE
A combined body moves in a direction neither cart initially had
A 2 kg cart moves east at 3 m/s and a 3 kg cart moves north at 4 m/s. They collide and stick on a smooth horizontal plane. External horizontal impulse is negligible. Find the final velocity vector and kinetic-energy loss.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions Focus: Conserve each momentum component in the same frame Learning goals: - Add momentum as a vector and interpret impulse as its change. - Assess external impulse before applying conservation. - Solve sticking and elastic one-dimensional impacts. - Use mass-weighted centre-of-mass position and velocity. - Resolve two-dimensional collisions and identify missing information. - Separate collision dynamics from later energy-conserving motion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
09 / Impulse, momentum and collisions
A special elastic collision has a useful geometric check
For two equal masses, with one initially at rest and an elastic collision in a plane, momentum conservation gives u = v₁ + v₂. Kinetic-energy conservation gives u² = v₁² + v₂². Squaring the vector momentum relation adds a cross term 2v₁·v₂. Comparing the two equations shows that the outgoing velocities are perpendicular when both are nonzero.
This right-angle result has specific conditions: equal masses, a stationary target in the chosen frame, conservation of total momentum and an elastic collision. It is not a rule for unequal masses, a moving target or an inelastic impact. The degenerate case where one outgoing speed is zero has no meaningful angle for that zero vector.
Use this relation as a check or a derived shortcut after identifying the conditions. In a general oblique collision, resolve along a stated line of impact and use the appropriate impulse and restitution model. A missing collision normal cannot be replaced by an assumed right angle.
WORKED EXAMPLE
Check both components and energy in an oblique collision
Two equal 1 kg particles collide on a smooth plane. Initially A moves at (5, 0) m/s and B is at rest. After collision A has velocity (4, 2) m/s. With negligible external impulse, find B’s velocity and determine whether these measured velocities are consistent with an elastic collision.
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STOP & THINK
When can nonzero outgoing velocities be inferred to be perpendicular in the standard two-body planar collision result?
- For every collision on a smooth surface
- For an elastic collision of equal masses with one initially at rest in that frame
- Whenever total momentum is zero
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions Focus: A special elastic collision has a useful geometric check Learning goals: - Add momentum as a vector and interpret impulse as its change. - Assess external impulse before applying conservation. - Solve sticking and elastic one-dimensional impacts. - Use mass-weighted centre-of-mass position and velocity. - Resolve two-dimensional collisions and identify missing information. - Separate collision dynamics from later energy-conserving motion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
10 / Impulse, momentum and collisions
Use different laws for the impact and the later motion
A collision followed by a rise, a spring compression or sliding on a rough surface has distinct stages. During a short inelastic impact, momentum may be approximately conserved while kinetic energy is not. During a later smooth rise, mechanical energy may be conserved while momentum changes under external gravity and support forces.
Write the state immediately before and immediately after the impact. Carry the post-impact velocity into the next stage. Equating the initial projectile kinetic energy directly to the final gravitational potential would incorrectly erase the energy converted during sticking.
For recoil or fragmentation, likewise separate the short internal event from subsequent flight. The combined momentum change during the event is set by external impulse. Afterwards, each part follows its own force law. A single conservation statement cannot be stretched across an interval whose system boundary or external influences have changed.
WORKED EXAMPLE
First stick, then climb
A 1 kg cart moving at 8 m/s sticks to a 3 kg cart at rest at the bottom of a smooth track that then rises. Neglect external horizontal impulse during the brief collision and all losses during the later climb. Use g = 10 m/s². Find the vertical rise of the combined carts before they momentarily stop.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Impulse, momentum and collisions Focus: Use different laws for the impact and the later motion Learning goals: - Add momentum as a vector and interpret impulse as its change. - Assess external impulse before applying conservation. - Solve sticking and elastic one-dimensional impacts. - Use mass-weighted centre-of-mass position and velocity. - Resolve two-dimensional collisions and identify missing information. - Separate collision dynamics from later energy-conserving motion. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- Momentum conservation depends on net external impulse over the chosen interval.
- Impulse is a force–time integral; work is a force–displacement integral.
- Sticking collisions generally convert relative translational energy into internal forms.
- The centre of mass responds to the external resultant.
- Two-dimensional momentum conservation requires a component equation for each axis.
- Use the correct law separately for each stage of a multi-stage event.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A 3 kg cart at +4 m/s sticks to a 1 kg cart at rest. External horizontal impulse is negligible. What is their common signed velocity?
Answer in m/s.
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SELF CHECK 2
A net impulse of 5 N·s acts over 0.25 s along +x. What is the average net force component?
Answer in N.
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SELF CHECK 3
A fixed-mass system has zero external resultant. Which centre-of-mass statement follows?
- It must be at rest
- Its velocity is constant
- It must lie inside solid material
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SELF CHECK 4
A container steadily receives moving particles. Why must its momentum balance be handled carefully?
- Momentum has ceased to be a vector
- The selected container system exchanges mass and momentum across its boundary
- Gravity always makes conservation impossible
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SELF CHECK 5
For a general planar elastic collision with two unknown final velocity vectors, do two momentum equations and one energy equation always determine all four components?
- Yes
- No; geometry or another collision condition may be needed
- Only if momentum is ignored
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
From a collision to a pendulum riseAdvanced · Free sample · Hints and guided reasoningTwo carts leave togetherIntermediate · Free sample · Hints and guided reasoningCombine two momentum directionsIntermediate · Hints and guided reasoningFollow the system, not either cartIntermediate · Hints and guided reasoningHow much does a triangular force pulse change velocity?Intermediate · Hints and guided reasoningMomentum survives the collision. Does kinetic energy?Advanced · Hints and guided reasoningA collision leaves less energy for the springAdvanced · Hints and guided reasoningA launcher recoilsIntermediate · Hints and guided reasoningAn elastic collision exchanges velocitiesIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Two carts stick together after colliding on a nearly frictionless track. Which conservation law gives their shared velocity?
Compare your reasoning
Total momentum is conserved over the collision if external impulse is negligible. Kinetic energy generally decreases; some becomes deformation, internal energy and sound.
Revisit the explanation →A few days later: change the situation
Two people initially at rest on frictionless ice push apart. If one has twice the mass, compare their speeds.
Compare your reasoning
Their momenta are equal in magnitude and opposite in direction, so the heavier person has half the speed. The centre of mass stays at rest, while internal energy can become kinetic energy.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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