Let's analyze the problem step-by-step. We need to find out which trial's cart has the greatest momentum at the bottom of the ramp.
Momentum ([tex]\( p \)[/tex]) is calculated using 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 at the bottom of the ramp.
We have four trials with the following data:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Trial} & \text{Mass of Cart (kg)} & \text{Velocity at Bottom (m/s)} \\
\hline
1 & 200 & 6.5 \\
2 & 220 & 5.0 \\
3 & 240 & 6.4 \\
4 & 260 & 4.8 \\
\hline
\end{array}
\][/tex]
Let's calculate the momentum for each trial:
1. Trial 1:
[tex]\[ p_1 = 200 \, \text{kg} \times 6.5 \, \text{m/s} = 1300 \, \text{kg} \cdot \text{m/s} \][/tex]
2. Trial 2:
[tex]\[ p_2 = 220 \, \text{kg} \times 5.0 \, \text{m/s} = 1100 \, \text{kg} \cdot \text{m/s} \][/tex]
3. Trial 3:
[tex]\[ p_3 = 240 \, \text{kg} \times 6.4 \, \text{m/s} = 1536 \, \text{kg} \cdot \text{m/s} \][/tex]
4. Trial 4:
[tex]\[ p_4 = 260 \, \text{kg} \times 4.8 \, \text{m/s} = 1248 \, \text{kg} \cdot \text{m/s} \][/tex]
Now we compare these momenta:
- Trial 1: [tex]\( 1300 \, \text{kg} \cdot \text{m/s} \)[/tex]
- Trial 2: [tex]\( 1100 \, \text{kg} \cdot \text{m/s} \)[/tex]
- Trial 3: [tex]\( 1536 \, \text{kg} \cdot \text{m/s} \)[/tex]
- Trial 4: [tex]\( 1248 \, \text{kg} \cdot \text{m/s} \)[/tex]
From these values, we see that Trial 3 has the greatest momentum: [tex]\( 1536 \, \text{kg} \cdot \text{m/s} \)[/tex].
Therefore, the answer is:
Trial 3, because this trial has a large mass and a large velocity.
