Question

part 1: momentum & collisions

1. define momentum. give two examples (high vs low momentum).

1. what happens to momentum if:

• velocity doubles?
• mass doubles?
• both double?

1. define impulse and explain why extending collision time (e.g., airbags, crumple zones) reduces force.

Explanation:

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<reasoning>

Define momentum and provide examples

\[

$$\begin{aligned} &p = mv \\ &\text{High momentum: A fast-moving freight train.} \\ &\text{Low momentum: A rolling tennis ball.} \end{aligned}$$

\]

Analyze proportional changes in momentum

\[

$$\begin{aligned} &p \propto v \implies \text{If velocity doubles, momentum doubles (2x).} \\ &p \propto m \implies \text{If mass doubles, momentum doubles (2x).} \\ &p \propto mv \implies \text{If both double, momentum quadruples (4x).} \end{aligned}$$

\]

Define impulse and analyze impact force reduction

\[

$$\begin{aligned} &J = F \Delta t = \Delta p \\ &F = \frac{\Delta p}{\Delta t} \\ &\Delta t \uparrow \implies F \downarrow \quad (\text{for a constant } \Delta p) \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

Momentum is the product of an object's mass and its velocity, represented by the formula \(p = mv\).

• High momentum example: A heavy freight train moving at high speed.
• Low momentum example: A slow-rolling tennis ball.

Question 2

• Velocity doubles: Momentum doubles (\(2p\)).
• Mass doubles: Momentum doubles (\(2p\)).
• Both double: Momentum quadruples (\(4p\)).

Question 3

Impulse is the product of the average force applied to an object and the time interval over which it acts, equal to the change in momentum (\(J = F\Delta t = \Delta p\)). Extending the collision time (\(\Delta t\)) reduces the impact force (\(F\)) because the required change in momentum (\(\Delta p\)) is spread over a longer duration, as shown by \(F = \frac{\Delta p}{\Delta t}\).
</answer>

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Answer:

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<reasoning>

Define momentum and provide examples

\[

$$\begin{aligned} &p = mv \\ &\text{High momentum: A fast-moving freight train.} \\ &\text{Low momentum: A rolling tennis ball.} \end{aligned}$$

\]

Analyze proportional changes in momentum

\[

$$\begin{aligned} &p \propto v \implies \text{If velocity doubles, momentum doubles (2x).} \\ &p \propto m \implies \text{If mass doubles, momentum doubles (2x).} \\ &p \propto mv \implies \text{If both double, momentum quadruples (4x).} \end{aligned}$$

\]

Define impulse and analyze impact force reduction

\[

$$\begin{aligned} &J = F \Delta t = \Delta p \\ &F = \frac{\Delta p}{\Delta t} \\ &\Delta t \uparrow \implies F \downarrow \quad (\text{for a constant } \Delta p) \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

Momentum is the product of an object's mass and its velocity, represented by the formula \(p = mv\).

• High momentum example: A heavy freight train moving at high speed.
• Low momentum example: A slow-rolling tennis ball.

Question 2

• Velocity doubles: Momentum doubles (\(2p\)).
• Mass doubles: Momentum doubles (\(2p\)).
• Both double: Momentum quadruples (\(4p\)).

Question 3

Impulse is the product of the average force applied to an object and the time interval over which it acts, equal to the change in momentum (\(J = F\Delta t = \Delta p\)). Extending the collision time (\(\Delta t\)) reduces the impact force (\(F\)) because the required change in momentum (\(\Delta p\)) is spread over a longer duration, as shown by \(F = \frac{\Delta p}{\Delta t}\).
</answer>

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