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 and final kinetic energy of the system and then find the difference between the two.
The initial kinetic energy of the car-truck system is the sum of the individual kinetic energies of the car and the truck.
The initial kinetic energy of the car is given by:
K1 = (1/2) * m * v1^2
The initial kinetic energy of the truck is given by:
K2 = (1/2) * (3m) * v2^2
Therefore, the initial kinetic energy of the car-truck system is:
K_initial = K1 + K2
After the collision, the car and the truck will move together with a common velocity v_final. Since they collide and couple together, their individual velocities will become the same.
To find the final velocity of the car-truck system, we can use the principle of conservation of linear momentum. The total momentum before the collision is equal to the total momentum after the collision.
The initial momentum of the car is:
P1_initial = m * v1
The initial momentum of the truck is:
P2_initial = (3m) * v2
The final momentum of the car-truck system is:
P_final = P1_final + P2_final = (4m) * v_final
Since total momentum is conserved, we have:
P_initial = P_final
m * v1 + (3m) * v2 = (4m) * v_final
Simplifying the equation, we find:
v_final = (m * v1 + 3m * v2) / (4m)
v_final = (v1 + 3v2) / 4
The final kinetic energy of the car-truck system is given by:
K_final = (1/2) * (4m) * v_final^2
Now, we can calculate the change in kinetic energy:
ΔK = K_final - K_initial
ΔK = (1/2) * (4m) * v_final^2 - [(1/2) * m * v1^2 + (1/2) * (3m) * v2^2]
Simplifying the equation, we find:
ΔK = (1/2) * (4m) * [(v1 + 3v2) / 4]^2 - [(1/2) * m * v1^2 + (1/2) * (3m) * v2^2]
Now, you can substitute the given values of v1, v2, and m into the equation to find the numerical value of the change in kinetic energy of the car-truck system in the collision.