Students are completing a lab in which they let a lab cart roll down a ramp. The students record the mass of the cart, the height of the ramp, and the velocity at the bottom of the ramp. The students then calculate the momentum of the cart at the bottom of the ramp.

\begin{tabular}{|c|c|c|c|}
\hline
Trial & \begin{tabular}{c}
Mass of \\
Cart (kg)
\end{tabular} & \begin{tabular}{c}
Height of Ramp \\
[tex]$(m)$[/tex]
\end{tabular} & \begin{tabular}{c}
Velocity at \\
Bottom (m/s)
\end{tabular} \\
\hline
1 & 200 & 2.0 & 6.5 \\
\hline
2 & 220 & 2.1 & 5.0 \\
\hline
3 & 240 & 1.5 & 6.4 \\
\hline
4 & 260 & 1.2 & 4.8 \\
\hline
\end{tabular}

Which trial's cart has the greatest momentum at the bottom of the ramp?

A. Trial 1, because this trial has the greatest velocity.
B. Trial 2, because this trial has the greatest ramp height.
C. Trial 3, because this trial has a large mass and a large velocity.
D. Trial 4, because this trial has the greatest mass.



Answer :

Let's work through the problem step-by-step to determine which trial's cart has the greatest momentum at the bottom of the ramp.

1. Identify the relevant quantities: The momentum of an object is given by the product of its mass and its velocity. Therefore, for each trial, we need to calculate the momentum using the formula:
[tex]\[ \text{momentum} = \text{mass} \times \text{velocity} \][/tex]

2. Calculate the momentum for each trial:

- Trial 1:
[tex]\[ \text{mass} = 200\, \text{kg}, \quad \text{velocity} = 6.5\, \text{m/s} \][/tex]
[tex]\[ \text{momentum} = 200 \, \text{kg} \times 6.5 \, \text{m/s} = 1300.0 \, \text{kg}\cdot\text{m/s} \][/tex]

- Trial 2:
[tex]\[ \text{mass} = 220\, \text{kg}, \quad \text{velocity} = 5.0\, \text{m/s} \][/tex]
[tex]\[ \text{momentum} = 220 \, \text{kg} \times 5.0 \, \text{m/s} = 1100.0 \, \text{kg}\cdot\text{m/s} \][/tex]

- Trial 3:
[tex]\[ \text{mass} = 240\, \text{kg}, \quad \text{velocity} = 6.4\, \text{m/s} \][/tex]
[tex]\[ \text{momentum} = 240 \, \text{kg} \times 6.4 \, \text{m/s} = 1536.0 \, \text{kg}\cdot\text{m/s} \][/tex]

- Trial 4:
[tex]\[ \text{mass} = 260\, \text{kg}, \quad \text{velocity} = 4.8\, \text{m/s} \][/tex]
[tex]\[ \text{momentum} = 260 \, \text{kg} \times 4.8 \, \text{m/s} = 1248.0 \, \text{kg}\cdot\text{m/s} \][/tex]

3. Compare the calculated momenta:

[tex]\[ \begin{align*} \text{Momentum for Trial 1} & = 1300.0\, \text{kg}\cdot\text{m/s} \\ \text{Momentum for Trial 2} & = 1100.0\, \text{kg}\cdot\text{m/s} \\ \text{Momentum for Trial 3} & = 1536.0 \, \text{kg}\cdot\text{m/s} \\ \text{Momentum for Trial 4} & = 1248.0 \, \text{kg}\cdot\text{m/s} \end{align*} \][/tex]

4. Determine which trial has the greatest momentum:
By comparing the momenta, we see that Trial 3 has the greatest momentum at [tex]\( 1536.0 \, \text{kg}\cdot\text{m/s} \)[/tex].

Therefore, the cart in Trial 3 has the greatest momentum at the bottom of the ramp.

The correct answer would be:
Trial 3, because this trial has a large mass and a large velocity.