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.
