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?

i cant seem to find the error but i still cant get the correct answer

You forgot the negative at the beginning in part B. Everything else is correct

assignment

To calculate the change in kinetic energy of the car-truck system in the collision, we need to find the initial and final kinetic energies and then subtract the initial kinetic energy from the final kinetic energy.

The initial kinetic energy of the car is given by the formula:
KE_car_initial = (1/2) * m * v1^2

The initial kinetic energy of the truck is given by:
KE_truck_initial = (1/2) * 3m * v2^2

Now, after the collision, the car and truck couple together and move with a final velocity. Let's denote the final velocity as Vf.

The final kinetic energy of the car-truck system can be calculated using the total mass of the car and truck, which is 4m since the truck has a mass of 3m and the car has a mass of m:
KE_final = (1/2) * (4m) * Vf^2

To find the change in kinetic energy, we subtract the initial kinetic energy from the final kinetic energy:
Change in KE = KE_final - (KE_car_initial + KE_truck_initial)

Substituting the above values, we get:
Change in KE = [(1/2) * (4m) * Vf^2] - [(1/2) * m * v1^2] - [(1/2) * 3m * v2^2]

We can simplify the equation further if we expand and combine like terms:
Change in KE = (2m * Vf^2) - (m * v1^2) - (1.5m * v2^2)

Therefore, the change in kinetic energy of the car-truck system in the collision is given by the equation:
Change in KE = 2m * Vf^2 - m * v1^2 - 1.5m * v2^2

CHECK MY ARITHMETIC !!!

It came out awful coincidental though.

Do it all over very carefully is all I can suggest.

initial momentum = m V1 + 3m V2

final momentum = 4 m V3
same final momentum as initial
so
V3 = (V1+3 V2)/4

initial Ke = .5 m( V1^2 + 3 V2^2)

final Ke = .5 (4m) (V3^2) = 2 m(V3^2) = 4 (.5m)V3^2

so change = .5m (V1^2+3V2^2 - 4 V3^2)
but
V3^2 = (V1^2 + 6V1V2 + 9V2^2)/16
4 V3^2 = (V1^2 + 6V1V2 + 9V2^2)/4
so change =
(1/2) m( (3/4)V1^2 -6/4 V1V2 +(3/4)V2^2)

= (m/8) (3 V1^2 - 6 V1V2 + 3 V2^2)

= (3 m/8)(V1-V2)^2