A 1000-kg car moving east at 80 km/h collides head-on with a 1500-kg car moving west at km/h, and the two cars stick together. (a) Which way does the wreckage move and with what initial speed? (b) How much KE is lost in the collision?

Question

A 1000-kg car moving east at 80 km/h collides head-on with a 1500-kg car moving west at  km/h, and the two cars stick together. (a) Which way does the wreckage move and with what initial speed? (b) How much KE is lost in the collision?

Bot · Answer

a) To find out which way the wreckage moves and with what initial speed, we can use the principle of conservation of momentum. The total momentum before the collision must be equal to the total momentum after the collision.

Before collision:

1000 kg car has momentum = (mass) * (speed)
= 1000 kg * 80 km/h
= 80000 kg km/h (east)

1500 kg car has momentum = (mass) * (speed)
= 1500 kg * 35 km/h
= 52500 kg km/h (west)

Because the directions are opposite, we can treat them as negative, so the net momentum before the collision is:

80000 kg km/h (east) - 52500 kg km/h (west)
= 27500 kg km/h (east)

After collision (since the two cars stick together, the total mass is 1000 kg + 1500 kg = 2500 kg):

Let V be the velocity of the wreckage after the collision. Then, the momentum after the collision is:
Momentum = (mass) * (velocity) = 2500 kg * V

By the conservation of momentum, we have:

27500 kg km/h (east) = 2500 kg * V

Now, we can solve for V:

V = 27500 kg km/h / 2500 kg = 11 km/h (east)

The wreckage moves towards the east with an initial speed of 11 km/h.

b) To find the kinetic energy (KE) lost in the collision, we can first find the KE before and after the collision and then find the difference between them.

Before collision:

KE of 1000 kg car = (1/2) * (mass) * (speed)^2
= 0.5 * 1000 kg * (80 km/h)^2
= 0.5 * 1000 * 6400 kg (km/h)^2
= 3200000 kg (km/h)^2

KE of 1500 kg car = (1/2) * (mass) * (speed)^2
= 0.5 * 1500 kg * (35 km/h)^2
= 0.5 * 1500 * 1225 kg (km/h)^2
= 918750 kg (km/h)^2

Total KE before collision = 3200000 kg (km/h)^2 + 918750 kg (km/h)^2 = 4118750 kg (km/h)^2

After collision:

Total mass = 2500 kg
Speed = 11 km/h

Total KE after collision = (1/2) * (mass) * (speed)^2
= 0.5 * 2500 kg * (11 km/h)^2
= 0.5 * 2500 * 121 kg (km/h)^2
= 151250 kg (km/h)^2

Now, we can find the KE lost in the collision:

KE lost = KE before collision - KE after collision
= 4118750 kg (km/h)^2 - 151250 kg (km/h)^2
= 3967500 kg (km/h)^2

Therefore, the kinetic energy lost in the collision is 3,967,500 kg (km/h)^2.