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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.

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

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 :

GinnyAnswer
To determine which trial's cart has the greatest momentum at the bottom of the ramp, we must calculate the momentum for each trial. The momentum [tex]\( p \)[/tex] of an object is given by the formula:

[tex]\[ p = m \times v \][/tex]

where [tex]\( m \)[/tex] is the mass of the cart and [tex]\( v \)[/tex] is the velocity of the cart at the bottom of the ramp.

Let's take a look at each trial step by step:

Trial 1:
- Mass [tex]\( m_1 = 200\ \text{kg} \)[/tex]
- Velocity [tex]\( v_1 = 6.5\ \text{m/s} \)[/tex]
- Momentum [tex]\( p_1 = 200 \times 6.5 = 1300.0\ \text{kg} \cdot \text{m/s} \)[/tex]

Trial 2:
- Mass [tex]\( m_2 = 220\ \text{kg} \)[/tex]
- Velocity [tex]\( v_2 = 5.0\ \text{m/s} \)[/tex]
- Momentum [tex]\( p_2 = 220 \times 5.0 = 1100.0\ \text{kg} \cdot \text{m/s} \)[/tex]

Trial 3:
- Mass [tex]\( m_3 = 240\ \text{kg} \)[/tex]
- Velocity [tex]\( v_3 = 6.4\ \text{m/s} \)[/tex]
- Momentum [tex]\( p_3 = 240 \times 6.4 = 1536.0\ \text{kg} \cdot \text{m/s} \)[/tex]

Trial 4:
- Mass [tex]\( m_4 = 260\ \text{kg} \)[/tex]
- Velocity [tex]\( v_4 = 4.8\ \text{m/s} \)[/tex]
- Momentum [tex]\( p_4 = 260 \times 4.8 = 1248.0\ \text{kg} \cdot \text{m/s} \)[/tex]

Now, comparing the calculated momenta:
- [tex]\( p_1 = 1300.0\ \text{kg} \cdot \text{m/s} \)[/tex]
- [tex]\( p_2 = 1100.0\ \text{kg} \cdot \text{m/s} \)[/tex]
- [tex]\( p_3 = 1536.0\ \text{kg} \cdot \text{m/s} \)[/tex]
- [tex]\( p_4 = 1248.0\ \text{kg} \cdot \text{m/s} \)[/tex]

Clearly, the momentum [tex]\( p_3 = 1536.0\ \text{kg} \cdot \text{m/s} \)[/tex] is the greatest.

Thus, the trial with the greatest momentum at the bottom of the ramp is Trial 3, because this trial has a large mass and a large velocity.

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