A 601-kg car stopped at an intersection is rear-ended by a 1660-kg truck moving with a speed of 14.5 m/s. If the car was in neutral and its brakes were off, so that the collision is approximately elastic, find the final speed of both vehicles after the collision.
To find the final speed of both vehicles after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision, provided no external forces act on the system.
The momentum (p) of an object is calculated by multiplying its mass (m) by its velocity (v). So, we can calculate the initial momentum (p_initial) for both the car and the truck and then find the final velocity (v_final) for each vehicle.
The initial momentum of the car (p_initial_car) is given by the mass of the car (m_car) times its initial velocity (v_initial_car), which is zero because the car is stopped:
p_initial_car = m_car * v_initial_car = 601 kg * 0 m/s = 0 kg·m/s
The initial momentum of the truck (p_initial_truck) is given by the mass of the truck (m_truck) times its initial velocity (v_initial_truck):
p_initial_truck = m_truck * v_initial_truck = 1660 kg * 14.5 m/s
To find the final velocities of both vehicles, we can equate the initial total momentum to the final total momentum:
p_initial_car + p_initial_truck = p_final_car + p_final_truck
Since the car is in neutral and its brakes are off, so that the collision is approximately elastic, we can assume that the final momentum of the truck and the car (p_final_car and p_final_truck) are equal to their initial momenta (p_initial_car and p_initial_truck):
p_initial_car + p_initial_truck = p_initial_car + p_initial_truck
Now we can solve for the final velocity of the car (v_final_car) and the final velocity of the truck (v_final_truck).
For the car:
p_initial_car = p_final_car
0 kg·m/s = m_car * v_final_car
Since the mass of the car (m_car) is given as 601 kg, we can solve for v_final_car:
v_final_car = 0 kg·m/s / 601 kg = 0 m/s
For the truck:
p_initial_truck = p_final_truck
1660 kg * 14.5 m/s = m_truck * v_final_truck
Since the mass of the truck (m_truck) is given as 1660 kg, we can solve for v_final_truck:
v_final_truck = (1660 kg * 14.5 m/s) / 1660 kg = 14.5 m/s
Therefore, the final speed of the car is 0 m/s, and the final speed of the truck is 14.5 m/s.