Momentum

A 2.0 kg trolley moves east at 3.0 m s⁻¹ and a 1.0 kg trolley moves west at 4.0 m s⁻¹. Take east as positive. What is the total momentum of the two-trolley system?

1. A

   +10 kg m s⁻¹

2. B

   +2.0 kg m s⁻¹

3. C

   −2.0 kg m s⁻¹

4. D

   +5.0 kg m s⁻¹

1. a

   Select and show Select the correct total momentum and show how direction is included.

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   Correct choiceB

   1. 1

      Selects option B: the system's total momentum is +2.0 kg m s⁻¹.

   2. 2

      Calculates eastward momentum as 2.0 × 3.0 = +6.0 kg m s⁻¹ and westward momentum as 1.0 × (−4.0) = −4.0 kg m s⁻¹.

   3. 3

      Adds signed momenta to obtain +2.0 kg m s⁻¹, meaning eastward.

A 0.20 kg ball approaches a bat at 10 m s⁻¹. Take the ball's initial direction as negative. A force sensor records the force on the ball; straight lines may be drawn between the data points.

1. a

   Determine Determine the impulse delivered to the ball from the area under the force–time graph.

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   1. 1

      Recognises impulse as the area under the force–time graph.

   2. 2

      Finds the total area, for example as four trapezia or one overall triangle: ½ × 0.08 × 100.

   3. 3

      Obtains impulse = +4.0 N s.

2. b

   Calculate Calculate the ball's velocity immediately after contact.

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   1. 1

      Calculates initial momentum = 0.20 × (−10) = −2.0 kg m s⁻¹.

   2. 2

      Uses final momentum = initial momentum + impulse = −2.0 + 4.0 = +2.0 kg m s⁻¹.

   3. 3

      Calculates final velocity = +2.0 ÷ 0.20 = +10 m s⁻¹.

3. c

   Explain Explain why using only the maximum force of 100 N would not determine the impulse.

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   1. 1

      Impulse depends on both force and the duration over which it acts.

   2. 2

      The force varies throughout contact, so the complete area rather than the peak value is required.

A 55 kg rider reaches a barrier at 6.0 m s⁻¹ and comes to rest. Barrier X gives an average stopping time of 0.12 s and can be reused after impact. Barrier Y gives 0.30 s but its foam cartridge must be replaced. Both stop the rider over the available distance.

1. a

   Calculate Calculate the magnitude of the average force on the rider for each barrier.

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   1. 1

      Calculates momentum change magnitude = 55 × 6.0 = 330 kg m s⁻¹.

   2. 2

      Calculates Barrier X average force = 330 ÷ 0.12 = 2750 N.

   3. 3

      Calculates Barrier Y average force = 330 ÷ 0.30 = 1100 N.

2. b

   Explain Explain physically why Barrier Y produces a smaller average force.

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   1. 1

      Both barriers produce the same momentum change for the same rider and approach speed.

   2. 2

      Barrier Y spreads that impulse over a longer time, so the average force is smaller.

3. c

   Evaluate Recommend a barrier and identify one further piece of evidence the park should obtain.

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   1. 1

      Makes a justified recommendation that weighs the much lower force from Y against replacement cost or waste, or prioritises another clearly stated constraint.

   2. 2

      Uses the numerical evidence accurately rather than making a general safety claim.

   3. 3

      Requests relevant evidence such as peak force, performance after rain, replacement time, life-cycle cost, or tests across rider masses and speeds.