Momentum and collision evidence

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Big idea: momentum and collision evidence

Key concept: Systems. Momentum equals mass times velocity and includes direction. Total momentum is conserved in a system with no resultant external force.
Related concepts: Models and evidence. Use a scientific explanation to make predictions, then test it against observations.
Global context: Scientific and technical innovation. Longer stopping times reduce average collision force for a given momentum change.

Momentum

Momentum = mass × velocity; p = mv
Units: kg m/s (mass in kg, velocity in m/s)
An object at rest (v = 0) has zero momentum
Momentum is a vector quantity – direction matters; can be positive or negative
Large momentum makes it difficult to change an object's direction

Conservation of Momentum

In a closed system (no external forces), total momentum before an event equals total momentum after
Total momentum before = Total momentum after
Total momentum is conserved in a system with no resultant external force.
For objects moving in opposite directions, momenta cancel (vector sum)

Conservation of momentum in a collision between two objects

Impulse

Impulse = force × time for which it acts; impulse = FΔt
Units: N s (newton seconds)
Impulse equals change in momentum: FΔt = Δp = mv – mu
Impulse is a vector – direction same as change in momentum

Force and Momentum

Newton's second law in terms of momentum: F = Δp / Δt
Force = rate of change of momentum (change in momentum per unit time)
A larger force or shorter contact time gives a larger change in momentum

A person holding an umbrella in rain versus hail, illustrating the difference in force and momentum change.

Worked Examples

Example 1: Tennis ball (0.06 kg) at 75 m/s has p = 4.5 kg m/s; brick (3 kg) at 1.5 m/s has same momentum
Example 2: Car (990 kg, 10 m/s) hits stationary van (4200 kg); after collision car moves at 2 m/s, van velocity = 1.9 m/s (conservation)
Example 3: Tennis ball (58 g) from 30 m/s left to 20 m/s right; impulse = –2.9 N s (to the right)
Example 4: Same Δp = 0.5 kg m/s; shorter contact time (0.1 s) gives larger force (5 N) than longer time (2 s gives 0.25 N)

Think like a scientist

Use a teacher-approved simulation to compare two-cart collisions with different initial speeds.
Comparison: the first cart's initial velocity. Outcome: the total momentum before and after collision.
Control: keep both cart masses constant. Explain why this makes the comparison fairer.
Evidence: Use a consistent method, repeated observations where appropriate and a table with labelled quantities and units. Keep unexpected results and investigate their cause.
Safety: Practical activities need teacher supervision and an appropriate risk assessment. Use the provided data or simulation where the investigation specifies it.
Inquiry task: State a testable question, predict the outcome using the science, then explain how your observations would support or challenge the prediction.

Evaluate the science

Longer stopping times reduce average collision force for a given momentum change.
Momentum conservation does not imply conservation of kinetic energy; some collisions transfer kinetic energy to other stores.
Evaluation task: Link your conclusion to evidence, identify a limitation and suggest a specific improvement. Distinguish a measured result from an explanation of its cause.

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Câu hỏi luyện tập

Xem trước miễn phí — 8 trên 52 câu hỏi. Đăng ký để xem tất cả.

1.What is the equation for momentum?
Easy
- Amomentum = mass × velocity
- Bmomentum = mass × acceleration
- Cmomentum = force × time
- Dmomentum = mass × speed
2.What are the units of momentum?
Easy
- Akg m/s
- Bkg m/s²
- CN s
- DN/m
3.What is the momentum of an object at rest?
Easy
- A0 kg m/s
- B1 kg m/s
- CIts mass in kg m/s
- DCannot be determined
4.A 5.0 kg ball is moving at 10 m/s. What is its momentum?
Easy
- A50 kg m/s
- B0.5 kg m/s
- C15 kg m/s
- D5 kg m/s
5.A 0.025 kg squash ball hits a wall at 44 m/s and rebounds at 42 m/s. What is the change in momentum? (Take initial direction as positive.)
Medium
- A-2.15 kg m/s
- B2.15 kg m/s
- C0.05 kg m/s
- D-0.05 kg m/s
6.What is the impulse on an object equal to?
Easy
- AThe change in momentum of the object
- BThe force multiplied by the distance moved
- CThe mass multiplied by the acceleration
- DThe velocity multiplied by the time
7.A 0.046 kg golf ball is hit, leaving the club at 65 m/s. The contact time is 5.0 × 10⁻⁴ s. What is the average force on the ball?
Medium
- A5980 N
- B2990 N
- C598 N
- D0.598 N
8.Which word best completes the sentence: 'Momentum is a property of a moving object making it difficult for it to change its ______.'
Easy
- Adirection
- Bmass
- Cspeed
- Dweight

Thêm câu hỏi

9.In a closed system, what is the total momentum before a collision compared to after?
Easy
- AEqual
- BGreater
- CLess
- DUnrelated
10.A 990 kg car moving at 10 m/s collides with a stationary 4200 kg van and they stick together. What is their combined velocity immediately after?
Medium
- A1.9 m/s
- B2.4 m/s
- C0.19 m/s
- D9.9 m/s
11.A 0.058 kg tennis ball moving at 30 m/s is struck and returns at 20 m/s in the opposite direction. What is the impulse on the ball? (Take initial direction as positive.)
Medium
- A-2.9 Ns
- B2.9 Ns
- C-0.58 Ns
- D0.58 Ns
12.A 0.05 kg golf ball is initially at rest. It is hit by a 0.5 kg putter moving at 0.2 m/s. After the hit, the putter stops. What is the velocity of the ball?
Medium
- A2.0 m/s
- B0.2 m/s
- C0.02 m/s
- D20 m/s
13.A 0.06 kg tennis ball moving at 75 m/s has the same momentum as a 3 kg brick moving at 1.5 m/s. Which statement about the forces they would exert on impact is correct?
Hard
- AThe tennis ball exerts a greater force because it has a higher speed.
- BThe brick exerts a greater force because it has a larger mass.
- CThey would exert similar forces because they have the same momentum.
- DThe force depends only on the time of impact, not on momentum.
14.In a collision between two objects in a closed system, the total momentum before the collision is 12 kg m/s to the right. After the collision, one object has a momentum of 5 kg m/s to the left. What is the momentum of the other object?
Medium
- A7 kg m/s to the right
- B17 kg m/s to the left
- C7 kg m/s to the left
- D17 kg m/s to the right
15.A tennis ball of mass 0.058 kg approaches a racket horizontally at 30 m/s to the left. It is hit and returns at 20 m/s to the right. What is the magnitude of the impulse exerted on the ball?
Medium
- A2.9 N s
- B0.58 N s
- C0.29 N s
- D5.8 N s
16.A ball hits a wall and bounces back. Which of the following best explains why the ball exerts a force on the wall?
Medium
- AThe ball's momentum changes, so by Newton's second law a force is required.
- BThe wall exerts an equal and opposite force on the ball.
- CThe ball's kinetic energy is conserved, so force is needed.
- DThe ball's mass increases during the collision.
17.For a collision in which the force on a ball varies with time, what does the area under a force-time graph represent?
Medium
- Athe impulse on the ball
- Bthe momentum of the ball
- Cthe kinetic energy of the ball
- Dthe acceleration of the ball
18.Which of the following is a vector quantity?
Easy
- Amomentum
- Bspeed
- Cmass
- Dkinetic energy
19.In an experiment to investigate conservation of momentum, two trolleys are placed on a frictionless track. Trolley A (mass 1.2 kg) moves at 0.5 m/s towards stationary trolley B (mass 0.8 kg). They collide and stick together. Which of the following describes the velocity of the combined trolleys after the collision?
Medium
- A0.3 m/s in the same direction as A initially
- B0.5 m/s in the same direction as A initially
- C0.3 m/s in the opposite direction to A initially
- D0.75 m/s in the same direction as A initially
20.A student measures the force on a ball during a collision using a force sensor. The force increases from 0 to a maximum and then decreases back to 0 over 0.02 s. The maximum force recorded is 150 N. Which of the following is the best estimate of the impulse on the ball?
Medium
- A1.5 N s
- B3.0 N s
- C0.75 N s
- D150 N s
21.In a laboratory experiment, a ball of mass 0.20 kg is dropped from a height onto a force sensor. It hits the sensor with speed 4.0 m/s and rebounds with speed 3.0 m/s. The contact time is 0.05 s. What is the average force exerted on the ball by the sensor?
Medium
- A28 N upward
- B4 N upward
- C28 N downward
- D4 N downward
22.Which of the following is a correct definition of impulse?
Easy
- AThe product of force and the time for which it acts
- BThe product of mass and velocity
- CThe rate of change of momentum
- DThe change in velocity divided by time
23.Two identical balls are dropped from the same height onto a hard floor. Ball A bounces back to nearly its original height, while ball B does not bounce. Which statement about the impulse on each ball from the floor is correct?
Medium
- AThe impulse on ball A is greater than the impulse on ball B
- BThe impulse on ball B is greater than the impulse on ball A
- CThe impulse on both balls is the same
- DThe impulse on ball A is zero
24.A student sets up an experiment to verify the conservation of momentum. Two trolleys of different masses are placed on a frictionless track. One trolley is pushed towards the other. After collision, the trolleys move together. Which measurement is NOT necessary to determine the velocity after collision?
Medium
- AThe time of contact during the collision
- BThe masses of both trolleys
- CThe initial velocity of the moving trolley
- DThe final velocity of the combined trolleys
25.The principle of conservation of momentum applies only when:
Easy
- Ano external forces act on the system
- Bthe collision is perfectly elastic
- Cthe objects stick together after collision
- Dthe masses of the objects are equal
26.
A force sensor records the force on a ball as a function of time during a collision, producing a force-time graph. Which quantity is obtained from the area under that graph?
Time (s) Force (N)
0.0 0
0.1 10
0.2 20
0.3 10
0.4 0
Easy
- AImpulse
- BMomentum
- CChange in velocity
- DAcceleration
27.A 0.50 kg ball moving at 2.0 m/s collides head-on with a stationary 1.0 kg ball. After the collision, the 0.50 kg ball moves backward at 0.5 m/s. What is the velocity of the 1.0 kg ball after the collision?
Medium
- A1.25 m/s forward
- B1.0 m/s forward
- C0.75 m/s forward
- D0.5 m/s forward
28.A student investigates the force exerted by a wall on a ball. The ball hits the wall perpendicularly and bounces back. Which of the following correctly describes the direction of the force on the ball?
Medium
- AAway from the wall, opposite to the ball's initial direction
- BToward the wall, in the ball's initial direction
- CParallel to the wall
- DThere is no force on the ball
29.A 4.0 kg trolley moving at 2.0 m/s collides with a stationary 8.0 kg trolley and they stick together. What is the velocity of the combined trolleys immediately after the collision?
Medium
- A0.67 m/s
- B1.0 m/s
- C2.0 m/s
- D0.33 m/s
30.In a closed system, the total momentum before an event is equal to the total momentum after the event. This is known as:
Easy
- Athe principle of conservation of momentum
- BNewton's first law
- CNewton's second law
- Dthe impulse-momentum theorem
31.Complete the sentence using the correct word.
Easy
Momentum is a property of a moving object that makes it difficult for it to change its ____.
32.A 0.060 kg tennis ball moving at 75 m/s has the same momentum as a 3.0 kg brick moving at 1.5 m/s. What is the momentum of the tennis ball?
Medium
33.A 0.025 kg squash ball hits a wall at 44 m/s and rebounds at 42 m/s. The ball is in contact with the wall for 0.1 s. What is the force exerted on the ball by the wall? (Take initial direction as positive.)
Medium
- A21.5 N
- B0.5 N
- C215 N
- D2.15 N
34.A tennis ball hits a racket twice with the same change in momentum of 0.5 kg m/s. The first contact time is 2 s and the second is 0.1 s. Which hit experiences the greater force?
Medium
- AThe second hit
- BThe first hit
- CBoth experience the same force
- DIt depends on the mass of the ball
35.What is the correct formula for momentum?
Easy
- Ap = m × v
- Bp = m × a
- Cp = m / v
- Dp = F × t
36.An object at rest has zero momentum.
Easy
True or false?
37.Which of the following statements about momentum are correct? (Select all that apply)
Medium
- AMomentum is a scalar quantity.
- BMomentum depends on both mass and velocity.
- CAn object with large momentum is difficult to stop.
- DMomentum can be negative.
38.In a closed system, the total momentum before a collision equals the total momentum after the collision.
Easy
True or false?
39.Impulse is equal to the change in momentum of an object.
Easy
True or false?
40.What are the units of impulse?
Easy
- AN s
- Bkg m/s²
- CJ
- DN/m
41.Use a teacher-approved simulation to compare two-cart collisions with different initial speeds. What is the independent variable?
Easy
- Athe total momentum before and after collision
- Bthe number of repeated trials
- Cthe first cart's initial velocity
- Dthe scientific explanation written afterwards
42.Use a teacher-approved simulation to compare two-cart collisions with different initial speeds. Which observation or quantity is the dependent variable?
Easy
- Athe first cart's initial velocity
- Bthe title given to the investigation
- Cthe number of references in the report
- Dthe total momentum before and after collision
43.Use a teacher-approved simulation to compare two-cart collisions with different initial speeds. Which decision best controls another factor in this comparison?
Medium
- Akeep both cart masses constant
- Bchange every condition for each trial
- Cuse a different measurement method for every result
- Drecord only results that agree with the prediction
44.Use a teacher-approved simulation to compare two-cart collisions with different initial speeds. Why should the class repeat measurements or check multiple comparable observations?
Medium
- ATo guarantee that every systematic error disappears
- BTo assess variation and judge how consistent the evidence is
- CTo make an unsupported prediction become correct
- DTo avoid reporting results that disagree with a model
45.A class is evaluating an application of momentum and collision evidence. Which statement identifies a relevant limitation?
Hard
- AAll scientific models reproduce every feature of the real world
- BOne favourable result proves that a technology has no disadvantages
- CMomentum conservation does not imply conservation of kinetic energy; some collisions transfer kinetic energy to other stores.
- DA conclusion needs no evidence when a scientific term is used
46.Use a teacher-approved simulation to compare two-cart collisions with different initial speeds. Which conclusion is most justified by a controlled comparison?
Hard
- AA claim that the result must hold in all possible conditions
- BA claim that alternative explanations never need testing
- CA claim based only on the most convenient observation
- DA conclusion restricted to the tested conditions, with uncertainty and limitations stated
47.Use a teacher-approved simulation to compare two-cart collisions with different initial speeds. Select the two practices that strengthen the validity of this comparison.
Medium
- Akeep both cart masses constant
- Bchange the measurement method whenever a result is unexpected
- Crecord the total momentum before and after collision using the same agreed procedure
- Dreport only observations that confirm the prediction
48.When evaluating evidence about momentum and collision evidence, select the two scientifically defensible statements.
Hard
- AOne result proves the explanation in every situation
- BMomentum conservation does not imply conservation of kinetic energy; some collisions transfer kinetic energy to other stores.
- CA conclusion should state the conditions and uncertainty of the evidence
- DModels never need revision when new evidence appears
49.Design a fair investigation for this question: Use a teacher-approved simulation to compare two-cart collisions with different initial speeds. Identify what to compare, what to record and one control.
Medium
50.Explain one use of the science in momentum and collision evidence and evaluate one limitation of the evidence or application.
Hard
51.A 2.0 kg cart moves at +3.0 m/s and sticks to a stationary 1.0 kg cart. With negligible external impulse, what is their shared velocity?
Hard
- A+2.0 m/s
- B+3.0 m/s
- C+6.0 m/s
- D-2.0 m/s
52.Two stopping methods produce the same momentum change, but one takes twice as long. How does its average stopping force compare?
Hard
- ATwice as large
- BHalf as large, assuming the same momentum change
- CUnchanged because time never affects force
- DFour times as large

Ghi chú ôn tập

## Big idea: momentum and collision evidence - **Key concept: Systems.** Momentum equals mass times velocity and includes direction. Total momentum is conserved in a system with no resultant external force. - **Related concepts: Models and evidence.** Use a scientific explanation to make predictions, then test it against observations. - **Global context: Scientific and technical innovation.** Longer stopping times reduce average collision force for a given momentum change. ## Momentum - **Momentum** = mass × velocity; **p = mv** - Units: kg m/s (mass in kg, velocity in m/s) - An object at rest (v = 0) has **zero momentum** - Momentum is a **vector** quantity – direction matters; can be positive or negative - Large momentum makes it difficult to change an object's direction ## Conservation of Momentum - In a **closed system** (no external forces), total momentum before an event equals total momentum after - **Total momentum before = Total momentum after** - Total momentum is conserved in a system with no resultant external force. - For objects moving in opposite directions, momenta cancel (vector sum) ## Impulse - **Impulse** = force × time for which it acts; **impulse = FΔt** - Units: N s (newton seconds) - Impulse equals **change in momentum**: **FΔt = Δp = mv – mu** - Impulse is a vector – direction same as change in momentum ## Force and Momentum - Newton's second law in terms of momentum: **F = Δp / Δt** - Force = **rate of change of momentum** (change in momentum per unit time) - A larger force or shorter contact time gives a larger change in momentum ## Worked Examples - Example 1: Tennis ball (0.06 kg) at 75 m/s has p = 4.5 kg m/s; brick (3 kg) at 1.5 m/s has same momentum - Example 2: Car (990 kg, 10 m/s) hits stationary van (4200 kg); after collision car moves at 2 m/s, van velocity = 1.9 m/s (conservation) - Example 3: Tennis ball (58 g) from 30 m/s left to 20 m/s right; impulse = –2.9 N s (to the right) - Example 4: Same Δp = 0.5 kg m/s; shorter contact time (0.1 s) gives larger force (5 N) than longer time (2 s gives 0.25 N) ## Think like a scientist - Use a teacher-approved simulation to compare two-cart collisions with different initial speeds. - **Comparison:** the first cart's initial velocity. **Outcome:** the total momentum before and after collision. - **Control:** keep both cart masses constant. Explain why this makes the comparison fairer. - **Evidence:** Use a consistent method, repeated observations where appropriate and a table with labelled quantities and units. Keep unexpected results and investigate their cause. - **Safety:** Practical activities need teacher supervision and an appropriate risk assessment. Use the provided data or simulation where the investigation specifies it. - **Inquiry task:** State a testable question, predict the outcome using the science, then explain how your observations would support or challenge the prediction. ## Evaluate the science - Longer stopping times reduce average collision force for a given momentum change. - Momentum conservation does not imply conservation of kinetic energy; some collisions transfer kinetic energy to other stores. - **Evaluation task:** Link your conclusion to evidence, identify a limitation and suggest a specific improvement. Distinguish a measured result from an explanation of its cause.

Conservation of Momentum: In a **closed system** (no external forces), total momentum before an event equals total momentum after
Impulse-Momentum Theorem: Relate impulse to change in momentum using FΔt = Δp, and calculate force or time from momentum change.
Momentum Calculation: Calculate momentum from mass and velocity, and understand momentum as a vector quantity.
Force as Rate of Change of Momentum: Use F = Δp/Δt to relate force to the rate of change of momentum, and compare forces from different impacts.
Designing scientific inquiries: Plan a fair comparison: Use a teacher-approved simulation to compare two-cart collisions with different initial speeds.
Evaluating evidence and impacts: Longer stopping times reduce average collision force for a given momentum change. Momentum conservation does not imply conservation of kinetic energy; some collisions transfer kinetic energy to other stores.
Momentum: **Momentum** = mass × velocity; **p = mv**
Impulse: **Impulse** = force × time for which it acts; **impulse = FΔt**
Force and Momentum: Newton's second law in terms of momentum: **F = Δp / Δt**
Worked Examples: Example 1: Tennis ball (0.06 kg) at 75 m/s has p = 4.5 kg m/s; brick (3 kg) at 1.5 m/s has same momentum
p = mv: **Momentum** = mass × velocity; **p = mv**
zero momentum: An object at rest (v = 0) has **zero momentum**
vector: Momentum is a **vector** quantity – direction matters; can be positive or negative
closed system: In a **closed system** (no external forces), total momentum before an event equals total momentum after
Total momentum before = Total momentum after: **Total momentum before = Total momentum after**
impulse = FΔt: **Impulse** = force × time for which it acts; **impulse = FΔt**

Thẻ ghi nhớ

What is the equation for momentum?

momentum = mass × velocity Momentum is defined as the product of mass and velocity. The correct equation is p = mv.

What are the units of momentum?

kg m/s Momentum = mass × velocity, so units are kg × m/s = kg m/s. Newton seconds (N s) are also equivalent but kg m/s is the standard unit.

What is the momentum of an object at rest?

0 kg m/s If velocity is zero, then momentum (mass × velocity) is zero.

A 5.0 kg ball is moving at 10 m/s. What is its momentum?

50 kg m/s Momentum p = mv = 5.0 kg × 10 m/s = 50 kg m/s.

A 0.025 kg squash ball hits a wall at 44 m/s and rebounds at 42 m/s. What is the change in momentum? (Take initial direction as positive.)

-2.15 kg m/s Initial momentum = 0.025 × 44 = 1.1 kg m/s. Final momentum = 0.025 × (-42) = -1.05 kg m/s. Change = final - initial = -1.05 - 1.1 = -2.15 kg m/s.

What is the impulse on an object equal to?

The change in momentum of the object Impulse = FΔt = Δp, so it equals the change in momentum.

A 0.046 kg golf ball is hit, leaving the club at 65 m/s. The contact time is 5.0 × 10⁻⁴ s. What is the average force on the ball?

5980 N Impulse = change in momentum = mv - 0 = 0.046 × 65 = 2.99 Ns. Force = impulse / time = 2.99 / (5.0 × 10⁻⁴) = 5980 N.

Which word best completes the sentence: 'Momentum is a property of a moving object making it difficult for it to change its ______.'

direction Momentum keeps an object moving in the same direction; it is difficult to change the direction of an object with large momentum.

In a closed system, what is the total momentum before a collision compared to after?

Equal The principle of conservation of momentum states that in a closed system, total momentum before equals total momentum after.

A 990 kg car moving at 10 m/s collides with a stationary 4200 kg van and they stick together. What is their combined velocity immediately after?

1.9 m/s Momentum before = 990 × 10 = 9900 kg m/s. After, total mass = 990 + 4200 = 5190 kg. Velocity = 9900 / 5190 ≈ 1.9 m/s.

A 0.058 kg tennis ball moving at 30 m/s is struck and returns at 20 m/s in the opposite direction. What is the impulse on the ball? (Take initial direction as positive.)

-2.9 Ns Impulse = m(v - u) = 0.058 × (-20 - 30) = 0.058 × (-50) = -2.9 Ns.

A 0.05 kg golf ball is initially at rest. It is hit by a 0.5 kg putter moving at 0.2 m/s. After the hit, the putter stops. What is the velocity of the ball?

2.0 m/s Conservation of momentum: initial momentum = putter momentum = 0.5 × 0.2 = 0.1 kg m/s. After, putter stops, so ball momentum = 0.1 = 0.05 × v ⇒ v = 2.0 m/s.

A 0.06 kg tennis ball moving at 75 m/s has the same momentum as a 3 kg brick moving at 1.5 m/s. Which statement about the forces they would exert on impact is correct?

They would exert similar forces because they have the same momentum. Momentum is mass × velocity. Both have the same momentum (4.5 kg m/s). On impact, the force depends on the change in momentum and contact time; if contact times are similar, the forces are similar. The worked example in the notes concludes this.

In a collision between two objects in a closed system, the total momentum before the collision is 12 kg m/s to the right. After the collision, one object has a momentum of 5 kg m/s to the left. What is the momentum of the other object?

17 kg m/s to the right Total momentum is conserved. Before: +12 kg m/s. After: one object has -5 kg m/s. So the other must have +17 kg m/s (to the right) to keep total +12.

A tennis ball of mass 0.058 kg approaches a racket horizontally at 30 m/s to the left. It is hit and returns at 20 m/s to the right. What is the magnitude of the impulse exerted on the ball?

2.9 N s Take left as positive: initial velocity = +30 m/s, final velocity = -20 m/s. Change in momentum = m(v-u) = 0.058(-20-30) = 0.058×(-50) = -2.9 kg m/s. Magnitude = 2.9 N s.

A ball hits a wall and bounces back. Which of the following best explains why the ball exerts a force on the wall?

The ball's momentum changes, so by Newton's second law a force is required. The change in momentum of the ball implies a force (F = Δp/Δt) is exerted on the ball, and by Newton's third law the ball exerts an equal and opposite force on the wall. The question asks why the ball exerts a force: it is because its momentum changes.

How would you design the inquiry for momentum and collision evidence?

Use a teacher-approved simulation to compare two-cart collisions with different initial speeds. Compare the first cart's initial velocity; record the total momentum before and after collision; keep both cart masses constant.

What limitation matters when applying momentum and collision evidence?

Momentum conservation does not imply conservation of kinetic energy; some collisions transfer kinetic energy to other stores.

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Time (s)	Force (N)
0.0	0
0.1	10
0.2	20
0.3	10
0.4	0