A 1,200-kg car runs into the rear of an 1,800-kg minivan that was moving at 6 m/s. The two vehicles are locked together after the collision and move with a speed of 12 m/s. How high above the ground would the locked- together vehicles have to be raised in order for their potential energyto equal the "lost" kinetic energy in the collision?
To find the height above the ground at which the locked-together vehicles need to be raised in order for their potential energy to equal the "lost" kinetic energy, we need to calculate the kinetic energy lost during the collision.
First, let's calculate the initial kinetic energy of the minivan before the collision:
Kinetic Energy (minivan) = (1/2) * mass (minivan) * velocity^2
= (1/2) * 1800 kg * (6 m/s)^2
= 36,000 Joules
Next, let's calculate the initial kinetic energy of the car before the collision:
Kinetic Energy (car) = (1/2) * mass (car) * velocity^2
= (1/2) * 1200 kg * (0 m/s)^2
= 0 Joules
The total initial kinetic energy before the collision is the sum of the kinetic energies of the minivan and the car:
Initial Kinetic Energy (total) = Kinetic Energy (minivan) + Kinetic Energy (car)
= 36,000 Joules + 0 Joules
= 36,000 Joules
Now, let's calculate the final kinetic energy after the collision. Since the vehicles are locked together, their combined mass is the sum of the masses of the minivan and the car.
Final Kinetic Energy = (1/2) * (Mass of minivan + Mass of car) * velocity^2
= (1/2) * (1800 kg + 1200 kg) * (12 m/s)^2
= (1/2) * 3000 kg * (12 m/s)^2
= 216,000 Joules
The kinetic energy lost during the collision is the difference between the initial and final kinetic energies:
Kinetic Energy Lost = Initial Kinetic Energy (total) - Final Kinetic Energy
= 36,000 Joules - 216,000 Joules
= -180,000 Joules (negative value indicates energy loss)
Since potential energy is given by mass * gravitational acceleration * height, we can set the potential energy equal to the kinetic energy lost and solve for the height above the ground:
Potential Energy = Kinetic Energy Lost
= -180,000 Joules
Mass (minivan + car) * 9.8 m/s^2 * height = -180,000 Joules
(3000 kg) * (9.8 m/s^2) * height = -180,000 Joules
Solving for height:
height = -180,000 Joules / (3000 kg * 9.8 m/s^2)
≈ -6.12 meters
The negative value indicates that the locked-together vehicles would need to be lowered below the ground by approximately 6.12 meters in order for their potential energy to equal the "lost" kinetic energy.