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?
To find the velocity of the truck after the elastic collision, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The formula for momentum is given by:
Momentum = Mass × Velocity
Let's assume the velocity of the truck after the collision is v2. Also, let's use the convention that negative velocity represents motion in the opposite direction.
Before the collision:
Momentum of car = Mass of car × Velocity of car
= (1.1 × 10^3 kg) × (+22.0 m/s) (since the car is moving in the positive direction)
= +2.42 × 10^4 kg m/s (taking right direction as positive)
Momentum of truck = Mass of truck × Velocity of truck
= (2.3 × 10^3 kg) × 0 m/s (since the truck is stationary)
= 0 kg m/s
Total momentum before the collision = Momentum of car + Momentum of truck
= +2.42 × 10^4 kg m/s + 0 kg m/s
= +2.42 × 10^4 kg m/s
After the collision:
Momentum of car after the collision = Mass of car × Velocity of car after the collision
= (1.1 × 10^3 kg) × (-11.0 m/s) (since the car is moving in the negative direction)
= -1.21 × 10^4 kg m/s (taking left direction as negative)
Momentum of truck after the collision = Mass of truck × Velocity of truck after the collision
= (2.3 × 10^3 kg) × v2
Total momentum after the collision = Momentum of car after the collision + Momentum of truck after the collision
= -1.21 × 10^4 kg m/s + (2.3 × 10^3 kg) × v2
According to the principle of conservation of momentum:
Total momentum before the collision = Total momentum after the collision
This can be written as:
(2.42 × 10^4 kg m/s) = (-1.21 × 10^4 kg m/s) + (2.3 × 10^3 kg) × v2
Now, we can solve this equation to find the velocity of the truck after the collision (v2).
To find the velocity of the truck after the elastic collision, we can use the law of conservation of momentum. According to this law, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum (p) of an object is given by the mass (m) multiplied by the velocity (v).
Initially, the momentum of the car is given by:
P_car_initial = m_car * v_car_initial
P_car_initial = (1.1 × 10^3 kg) * (22.0 m/s)
P_car_initial = 24,200 kg·m/s
Since the truck is stationary before the collision, its initial velocity is 0m/s. Therefore, the initial momentum of the truck is zero.
P_truck_initial = m_truck * v_truck_initial
P_truck_initial = (2.3 × 10^3 kg) * (0 m/s)
P_truck_initial = 0 kg·m/s
After the collision, the momentum of the car is given by:
P_car_final = m_car * v_car_final
P_car_final = (1.1 × 10^3 kg) * (-11.0 m/s)
P_car_final = -12,100 kg·m/s
To find the momentum of the truck after the collision, we can use the fact that the total momentum before and after the collision should be equal.
Total momentum before collision = Total momentum after collision
P_car_initial + P_truck_initial = P_car_final + P_truck_final
24,200 kg·m/s + 0 kg·m/s = -12,100 kg·m/s + P_truck_final
P_truck_final = 24,200 kg·m/s + 0 kg·m/s - (-12,100 kg·m/s)
P_truck_final = 24,200 kg·m/s + 12,100 kg·m/s
P_truck_final = 36,300 kg·m/s
Finally, we can find the velocity of the truck by dividing its final momentum by its mass:
P_truck_final = m_truck * v_truck_final
36,300 kg·m/s = (2.3 × 10^3 kg) * v_truck_final
v_truck_final = 36,300 kg·m/s / (2.3 × 10^3 kg)
v_truck_final ≈ 15.78 m/s
Therefore, the velocity of the truck after the elastic collision is approximately +15.78 m/s.