Question
Select the correct answer.
A car with a mass of 1.1 × 103 kilograms hits a stationary truck with a mass of 2.3 × 103 kilograms from the rear end. The initial velocity of the car is +22.0 meters/second. After the collision the velocity of the car is -11.0 meters/second. What is the velocity of the truck after this elastic collision?
A.
+13 meters/second
B.
+20 meters/second
C.
+15.7 meters/second
D.
+7.5 meters/second
E.
+9.8 meters/second
A car with a mass of 1.1 × 103 kilograms hits a stationary truck with a mass of 2.3 × 103 kilograms from the rear end. The initial velocity of the car is +22.0 meters/second. After the collision the velocity of the car is -11.0 meters/second. What is the velocity of the truck after this elastic collision?
A.
+13 meters/second
B.
+20 meters/second
C.
+15.7 meters/second
D.
+7.5 meters/second
E.
+9.8 meters/second
Answers
GPT-4o mini
To solve for the velocity of the truck after the collision, we can use the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.
The momentum before the collision is the sum of the momentum of the car and the truck:
\[
p_{\text{initial}} = m_{\text{car}} \cdot v_{\text{car, initial}} + m_{\text{truck}} \cdot v_{\text{truck, initial}}.
\]
Given:
- Mass of the car, \( m_{\text{car}} = 1.1 \times 10^3 \, \text{kg} \)
- Velocity of the car before the collision, \( v_{\text{car, initial}} = +22.0 \, \text{m/s} \)
- Mass of the truck, \( m_{\text{truck}} = 2.3 \times 10^3 \, \text{kg} \)
- Velocity of the truck before the collision, \( v_{\text{truck, initial}} = 0 \, \text{m/s} \)
Calculating the initial momentum:
\[
p_{\text{initial}} = (1.1 \times 10^3 \, \text{kg}) \cdot (22.0 \, \text{m/s}) + (2.3 \times 10^3 \, \text{kg}) \cdot (0 \, \text{m/s}),
\]
\[
p_{\text{initial}} = (1.1 \times 10^3 \times 22.0) + 0 = 24200 \, \text{kg m/s}.
\]
After the collision, the velocities are:
- Velocity of the car, \( v_{\text{car, final}} = -11.0 \, \text{m/s} \)
- Velocity of the truck after the collision is unknown, let’s denote it as \( v_{\text{truck, final}} \).
The total momentum after the collision is:
\[
p_{\text{final}} = m_{\text{car}} \cdot v_{\text{car, final}} + m_{\text{truck}} \cdot v_{\text{truck, final}}.
\]
Setting the initial momentum equal to the final momentum:
\[
24200 = (1.1 \times 10^3 \cdot (-11.0)) + (2.3 \times 10^3 \cdot v_{\text{truck, final}}).
\]
Calculate the momenta:
\[
24200 = (-12100) + (2.3 \times 10^3 \cdot v_{\text{truck, final}}).
\]
Adding \( 12100 \) to both sides:
\[
24200 + 12100 = 2.3 \times 10^3 \cdot v_{\text{truck, final}},
\]
\[
36300 = 2.3 \times 10^3 \cdot v_{\text{truck, final}}.
\]
Now, solve for \( v_{\text{truck, final}} \):
\[
v_{\text{truck, final}} = \frac{36300}{2.3 \times 10^3} \approx 15.8 \, \text{m/s}.
\]
Looking at the options, the closest available choice to our computed \( 15.8 \, \text{m/s} \) is:
**C. +15.7 meters/second**
The momentum before the collision is the sum of the momentum of the car and the truck:
\[
p_{\text{initial}} = m_{\text{car}} \cdot v_{\text{car, initial}} + m_{\text{truck}} \cdot v_{\text{truck, initial}}.
\]
Given:
- Mass of the car, \( m_{\text{car}} = 1.1 \times 10^3 \, \text{kg} \)
- Velocity of the car before the collision, \( v_{\text{car, initial}} = +22.0 \, \text{m/s} \)
- Mass of the truck, \( m_{\text{truck}} = 2.3 \times 10^3 \, \text{kg} \)
- Velocity of the truck before the collision, \( v_{\text{truck, initial}} = 0 \, \text{m/s} \)
Calculating the initial momentum:
\[
p_{\text{initial}} = (1.1 \times 10^3 \, \text{kg}) \cdot (22.0 \, \text{m/s}) + (2.3 \times 10^3 \, \text{kg}) \cdot (0 \, \text{m/s}),
\]
\[
p_{\text{initial}} = (1.1 \times 10^3 \times 22.0) + 0 = 24200 \, \text{kg m/s}.
\]
After the collision, the velocities are:
- Velocity of the car, \( v_{\text{car, final}} = -11.0 \, \text{m/s} \)
- Velocity of the truck after the collision is unknown, let’s denote it as \( v_{\text{truck, final}} \).
The total momentum after the collision is:
\[
p_{\text{final}} = m_{\text{car}} \cdot v_{\text{car, final}} + m_{\text{truck}} \cdot v_{\text{truck, final}}.
\]
Setting the initial momentum equal to the final momentum:
\[
24200 = (1.1 \times 10^3 \cdot (-11.0)) + (2.3 \times 10^3 \cdot v_{\text{truck, final}}).
\]
Calculate the momenta:
\[
24200 = (-12100) + (2.3 \times 10^3 \cdot v_{\text{truck, final}}).
\]
Adding \( 12100 \) to both sides:
\[
24200 + 12100 = 2.3 \times 10^3 \cdot v_{\text{truck, final}},
\]
\[
36300 = 2.3 \times 10^3 \cdot v_{\text{truck, final}}.
\]
Now, solve for \( v_{\text{truck, final}} \):
\[
v_{\text{truck, final}} = \frac{36300}{2.3 \times 10^3} \approx 15.8 \, \text{m/s}.
\]
Looking at the options, the closest available choice to our computed \( 15.8 \, \text{m/s} \) is:
**C. +15.7 meters/second**