Unit 4 · Topic 4.2 Beta

Change in Momentum and Impulse

Impulse J = F_avg Δt is the area under a force-time graph, and it equals the change in momentum.

Practice 1: Creating RepresentationsPractice 2: Mathematical RoutinesPractice 3: Scientific Questioning and Argumentation

Share to Google Classroom (opens in a new tab)

Question set for this topic

Part 1 · Hook

Why this matters

Airbags, gym mats and the crumple zones in cars do not change how much a body must slow down in a crash. They change how long the slowing takes. Impulse explains why stretching out the time is all it takes to turn a bone-breaking force into a survivable one.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Momentum is

  1. mass times velocity
  2. mass times acceleration
  3. force times distance
  4. ½mv²
Show the answer

p = mv.

  • Correct: mass times velocity:
  • mass times acceleration:
  • force times distance:
  • ½mv²:

2. On a velocity-time graph, the area under the curve gives the

  1. displacement
  2. acceleration
  3. force
  4. speed
Show the answer

Area under v-t is Δx; slope is acceleration.

  • Correct: displacement:
  • acceleration:
  • force:
  • speed:

3. Newton's second law for an object of constant mass is

  1. F_net = ma
  2. F_net = mv
  3. F_net = m/a
  4. F_net = ½ma²
Show the answer

The net force equals mass times acceleration.

  • Correct: F_net = ma:
  • F_net = mv:
  • F_net = m/a:
  • F_net = ½ma²:

Part 4 · See it

See it first

A 0.20 kg ball moves left at 4.0 m/s toward a wall, is pushed right by the wall during contact, and leaves right at 2.0 m/s. A triangular force-time graph peaks at 600 N at 2 ms and ends at 4 ms; its shaded area, 1.2 N·s, is the impulse.
A ball bounces off a wall. The area under the force-time graph is the impulse, equal to the ball's change in momentum. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. A force acts on an object over a time intervalit delivers an impulse J = F_avg Δt, the area under the force-time graph, in N·s
  2. The impulse acts on the objectthe object's momentum changes by exactly that amount: J = Δp
  3. The change in momentum is fixed (stop from a given speed)a longer contact time means a smaller average force
  4. Momentum is graphed against timethe slope of the p-t graph is the net force: F_net = Δp/Δt

Part 6 · Key ideas

Key ideas

  • Impulse J = F_avg Δt, a vector along the force, in N·s (= kg·m/s). It is the area under a force-time graph.
  • The impulse-momentum theorem: J_net = Δp = m v_f − m v_i. Keep the signs: a bounce reverses v.
  • The same Δp over a longer time needs a smaller force: airbags, mats, bent knees.
  • F_net = Δp/Δt, the slope of a momentum-time graph. For constant mass this is Newton's second law, F_net = ma.

Part 7 · Misconception

A common mistake

The wrong idea: Momentum is force times time, so p = Ft.

What actually happens: Force times time is the impulse, which equals the change in momentum, Δp = F_avg Δt. Momentum itself is mv.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Diagram

A ball bounces off a wall

A 0.20 kg ball bounces off a wallBeforevi = 4.0 m/sDuring contactFN, wall on ballAftervf = 2.0 m/sTime (ms)Force of wall on ball (N)012340300600

The ball has a mass of 0.20 kg. Take right as positive.

1. Use the graph to find the impulse the wall exerts on the ball. Give the value and its unit.

Type a number and its unit.

Show the answer

Impulse = area under F-t = ½(0.0040 s)(600 N) = 1.2 N·s, to the right.

  • Answer: 1.2 N·s

2. What is the ball's change in momentum, in kg·m/s? Include the sign.

Type a number in kg·m/s.

Show the answer

Δp = m v_f − m v_i = (0.20)(+2.0) − (0.20)(−4.0) = +1.2 kg·m/s, matching the impulse: J = Δp.

  • Answer: 1.2 kg·m/s

3. What is the average force of the wall on the ball during the 4.0 ms contact, in newtons?

Type a number in N.

Show the answer

F_avg = J/Δt = 1.2 N·s / 0.0040 s = 3.0 × 10² N, half the peak, as expected for a triangle.

  • Answer: 300 N

4. A softer ball with the same mass and the same speeds before and after stays in contact for 8.0 ms instead. How does the wall's force compare?

  1. Same impulse, half the average force
  2. Twice the impulse, same average force
  3. Same impulse, same average force
  4. Half the impulse, half the average force
Show the answer

J = F_avg Δt = Δp. Δp is the same, so doubling Δt halves F_avg. This is how airbags and helmets work.

  • Correct: Same impulse, half the average force: Right: Δp is unchanged, so J is the same, spread over twice the time.
  • Twice the impulse, same average force: The impulse equals Δp, which has not changed.
  • Same impulse, same average force: Spreading the same impulse over more time lowers the average force.
  • Half the impulse, half the average force: The impulse is fixed by the change in momentum, 1.2 N·s.

5. On a graph of force against time, what represents the impulse?

  1. The slope of the curve
  2. The highest force on the graph
  3. The area under the curve
  4. The time at which the force is largest
Show the answer

Impulse J = area under F against t. On a p-t graph, the slope is the net force.

  • The slope of the curve: The slope is a rate of change of force, not an impulse.
  • The highest force on the graph: A peak force says nothing about how long it acted.
  • Correct: The area under the curve: Right: impulse is the sum of F Δt over the interval.
  • The time at which the force is largest: Impulse needs force and time together.

6. An egg dropped onto a pillow is compared with the same egg dropped from the same height onto a hard floor (where it stops sooner). Predict how each pillow value compares with the floor value.

VariableChange
The egg's change in momentum—
The impulse on the egg—
The average force on the egg—
Show the answer

J = Δp is the same; F_avg = J/Δt is smaller on the pillow because the stop takes longer.

  • The egg's change in momentum: no change. It arrives at the same speed and stops in both cases.
  • The impulse on the egg: no change. Impulse equals the change in momentum.
  • The average force on the egg: decreases. The same impulse is spread over a longer time on the pillow.

Part 9 · Summary

Summary

Impulse J = F_avg Δt is the area under a force-time graph, and it equals the change in momentum. A fixed change in momentum spread over more time needs less force. The slope of a momentum-time graph is the net force.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections