AP Physics 1 4.3: Conservation of Linear Momentum

Analyze a collection of objects as a system and use center-of-mass motion and momentum conservation when the net external force is zero.

[Maximum number: 4]

A student has a cart of mass $m_{\mathrm{c}}$ and a block of mass $\frac{1}{5} m_{\mathrm{c}}$ , as shown in Figure 1.

- At time t=0, the cart is moving to the right across a horizontal surface with constant

speed $v_{\mathrm{c}}$ , and the student releases the block from rest.

- At $t=t_{1}$ , the block collides with and sticks to the top of the cart. The block does not slide

on the cart.

- At $t=t_{2}$ , the block-cart system continues to move to the right with constant speed $v_{f}$ .

Figure 1

(a)

A.

[ 4 ]

(i)

On the axes shown in Figure 2, sketch a graph of the magnitude $p_{x}$ of the x-component

of the momentum of the block-cart system as a function of time t from t=0 until $t>t_{2}$ .

Figure 2

[ 2 ]

A (i) For sketching one of the following:

- A constant $p_{x}$ for $t<t_{1}$

- A constant $p_{x}$ for $t>t_{1}$

For sketching a line that demonstrates momentum is constant

Point A2

Accept one of the following:

- A continuous, nonzero, horizontal line from t=0 to $t=t_{2}$

- A continuous, nonzero, horizontal line from t=0 to $t>t_{2}$

Example Response

Official scoring-guideline example response

(ii)

Derive an expression for the speed $v_{f}$ of the block-cart system after time $t=t_{2}$ in terms

of $m_{\mathrm{c}}, v_{\mathrm{c}}$ , and physical constants, as appropriate. Begin your derivation by writing a

fundamental physics principle or an equation from the reference information.

[ 2 ]

(ii) For including a conservation of momentum equation

Point A3

Scoring Note: Part A (ii) and part A (iii) may be scored together, if necessary.

For setting $m_{\mathrm{c}} v_{\mathrm{c}}$ equal to the momentum of the block-cart system after the collision

Point A4

Scoring Note: A correct, isolated, final expression of $v_{f}=\frac{5}{6} v_{\mathrm{c}}$ earns points A3 and

A4.

Example Response

$\begin{aligned} & p_{i}=p_{f} \\ & m_{\mathrm{c}} v_{\mathrm{c}}=\left(m_{\mathrm{c}}+m_{\mathrm{b}}\right) v_{f}=\left(m_{\mathrm{c}}+\frac{1}{5} m_{\mathrm{c}}\right) v_{f} \\ & v_{f}=\frac{5}{6} v_{\mathrm{c}} \end{aligned}$

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4.3.A—Describe the behavior of a system using conservation of linear momentum question 1

Question (a)

Question (i)

Question (ii)