A car of mass m moving at a speed v1 collides and couples with the back of a truck of mass 3m moving initially in the same direction as the car at a lower speed v2. (Use any variable or symbol stated above as necessary.)
(b) What is the change in kinetic energy of the car–truck system in the collision?
To find the change in kinetic energy of the car-truck system in the collision, we need to calculate the initial kinetic energy and the final kinetic energy of the system.
The initial kinetic energy of the car-truck system is the sum of the kinetic energies of the car and the truck. The final kinetic energy is the total kinetic energy after the collision, where the car and truck are coupled together.
1. Initial kinetic energy:
The kinetic energy of the car is given by: (1/2) * m * v1^2
The kinetic energy of the truck is given by: (1/2) * 3m * v2^2
So, the initial kinetic energy of the car-truck system is:
KE_initial = (1/2) * m * v1^2 + (1/2) * 3m * v2^2
2. Final kinetic energy:
After the collision, the car and truck are coupled together, so their masses add up. The speed of the coupled system will be given by the conservation of momentum equation:
(m * v1) + (3m * v2) = (4m * v_f)
Simplifying the equation, we get:
v_f = (m * v1 + 3m * v2) / (4m) = (v1 + 3v2) / 4
The final kinetic energy of the car-truck system is:
KE_final = (1/2) * (4m) * v_f^2 = (1/2) * (4m) * ([(v1 + 3v2) / 4])^2 = (v1 + 3v2)^2 / 8
3. Change in kinetic energy:
The change in kinetic energy is defined as the difference between the final and initial kinetic energies:
Change in KE = KE_final - KE_initial = (v1 + 3v2)^2 / 8 - [(1/2) * m * v1^2 + (1/2) * 3m * v2^2]
Simplifying further, we get:
Change in KE = (v1 + 3v2)^2 / 8 - [(1/2) * m * v1^2 + (3/2) * m * v2^2]
This is the expression for the change in kinetic energy of the car-truck system in the collision.
To determine the change in kinetic energy of the car-truck system in the collision, we need to calculate the initial kinetic energy and the final kinetic energy of the system and find the difference between them.
The kinetic energy of an object is given by the equation:
KE = 0.5 * mass * velocity^2
Let's calculate the initial kinetic energy of the car and the truck separately:
Initial kinetic energy of the car:
KE_car_initial = 0.5 * m * v1^2
Initial kinetic energy of the truck:
KE_truck_initial = 0.5 * 3m * v2^2 [Since the truck has 3 times the mass of the car]
Now, the car and the truck collide and couple together. After the collision, they move with a common final velocity, let's call it v_final.
The final kinetic energy of the car-truck system can be calculated as:
KE_final = 0.5 * (m + 3m) * v_final^2 [Since the total mass of the system is m + 3m]
The change in kinetic energy of the car-truck system is given by the difference between the initial and final kinetic energy:
Change in kinetic energy = KE_final - (KE_car_initial + KE_truck_initial)
Substituting the expressions for initial kinetic energy and final kinetic energy:
Change in kinetic energy = (0.5 * (m + 3m) * v_final^2) - (0.5 * m * v1^2) - (0.5 * 3m * v2^2)
Now you can simplify and solve the equation to get the final answer.
Note: To calculate the final velocity v_final, you will also need additional information about the collision, such as whether it is an elastic or inelastic collision, or any other relevant details.