Unit 4.4 | Momentum and Energy Together
- 16 minutes
- 2 formulas
- 22 questions
- 3 videos
The hardest momentum problems chain two conservation laws. This final lesson shows how to split them at the collision and pick the right law for each stage.
Unit 4 Breakdown
In this lesson:
- Split a multi-step problem at the collision
- Know which conservation law holds in each stage, and why
- The 4 problem types: collide then swing, collide then slide, collide then squeeze a spring, and energy then collide
Introduction
This lesson covers problems that need momentum and energy together: the multi-step problems that show up in almost every AP free-response section.
You have two conservation laws now: energy (Unit 3.3) and momentum (Unit 4.2). The hardest problems chain them, one after the other. The skill is knowing which one to use when.
Split It at the Collision
- Before the collision: if the object slides, swings or is launched first, use the energy framework to find its speed just before the hit.
- The collision: momentum only, [katex] \sum p_i = \sum p_f [/katex]. Kinetic energy is lost unless the collision is elastic.
- After the collision: energy again, to find a height, a distance or a compression.
Never use conservation of energy across a collision. It is the single most common mistake on these problems.
| Problem type | The clue | Stages |
| 1. Collide, then swing | bullet into a hanging block | momentum, then energy |
| 2. Collide, then slide | “slide a distance before stopping” | momentum, then energy with friction |
| 3. Collide, then squeeze a spring | a spring or ramp after the hit | momentum, then energy |
| 4. Energy, then collide | a ramp, swing or spring before the hit | energy, then momentum (then often energy) |
Problem Type 1: Collide, then swing (the ballistic pendulum)
How to spot it: A bullet or dart fired into a hanging block that swings up.
Setup: Momentum for the hit: [katex] mv_0 = (m + M)v’ [/katex]. Energy for the swing: [katex] \frac{1}{2}(m + M)v’^2 = (m + M)gh [/katex].
A [katex] 0.01\,\text{kg} [/katex] bullet at [katex] 400\,\text{m/s} [/katex] embeds in a [katex] 2\,\text{kg} [/katex] hanging block. How high does the block swing?
Collision: [katex] (0.01)(400) = (2.01)v’ \rightarrow v’ \approx 1.99 \, m/s [/katex].
Swing: [katex] h = \frac{v’^2}{2g} = \frac{(1.99)^2}{2(9.8)} \approx 0.20 \, m [/katex].
AP trap: Setting the bullet’s kinetic energy equal to the block’s final potential energy. Energy was lost in the collision, so that skips a stage and gives a height about 200 times too big.
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Problem Type 2: Collide, then slide to a stop
How to spot it: The objects stick and then “slide a distance before coming to rest” on a rough surface.
Setup: Work backward. Energy with friction for the slide: [katex] \frac{1}{2}(m_1 + m_2)v’^2 = \mu (m_1 + m_2)g d [/katex], so [katex] v’ = \sqrt{2\mu g d} [/katex]. Then momentum for the hit.
A [katex] 1500\,\text{kg} [/katex] car hits a stopped [katex] 1000\,\text{kg} [/katex] car. They lock together and skid [katex] 5\,\text{m} [/katex] before stopping. The coefficient of friction is 0.6. How fast was the first car going?
Slide: [katex] v’ = \sqrt{2(0.6)(9.8)(5)} \approx 7.67 \, m/s [/katex].
Collision: [katex] (1500)v_0 = (2500)(7.67) \rightarrow v_0 \approx 12.8 \, m/s [/katex].
AP trap: Using the mass of the first car only in the friction stage. After the hit, both cars slide together.
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Problem Type 3: Collide, then squeeze a spring or climb a ramp
How to spot it: After the hit, the objects run into a spring or roll up a ramp.
Setup: Momentum for the hit, then [katex] \frac{1}{2}(m_1 + m_2)v’^2 = \frac{1}{2}kx^2 [/katex] (or [katex] = (m_1 + m_2)gh [/katex] for a ramp).
A [katex] 1\,\text{kg} [/katex] block at [katex] 6\,\text{m/s} [/katex] hits and sticks to a [katex] 2\,\text{kg} [/katex] block at rest. The pair then compresses a spring with [katex] k = 300 \, N/m [/katex]. What is the maximum compression?
Collision: [katex] (1)(6) = (3)v’ \rightarrow v’ = 2 \, m/s [/katex].
Spring: [katex] \frac{1}{2}(3)(2)^2 = \frac{1}{2}(300)x^2 \rightarrow x = 0.2 \, m [/katex].
AP trap: Answers in terms of the starting kinetic energy. If a block sticks to one 3 times heavier, only a quarter of its kinetic energy survives the hit, and only that quarter reaches the spring.
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Problem Type 4: Energy first, then collide
How to spot it: The object slides down a ramp, swings down, or is launched by a spring before it hits something.
Setup: Run the stages in order: energy to get the speed just before the hit, momentum through the hit, then energy again if the problem keeps going.
▶ Watch: a worked energy-then-momentum problem (starts at 7:20).
AP trap: Starting the momentum equation with the height instead of the speed. Convert the height to a speed with [katex] v = \sqrt{2gh} [/katex] first.
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Unit 4 Recap
Review the whole unit in the Unit 4 Speed Recap, or watch the momentum section of my speed review:
Ready for the whole unit? Take the unit review:
Unit 5 Preview
Next up is Unit 5: Rotation. Everything you know about motion, forces and momentum gets a rotational twin: angle, angular velocity, torque and rotational inertia.
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This article was written by Jason