A automobile travels at a speed of northward along a street, and a 7000-N sports car travels at a speed of eastward along an intersecting street. (a) If neither driver brakes and the cars collide at the intersection and lock bumpers, what will the velocity of the cars be immediately after the collision? (b) What percentage of the initial kinetic energy will be lost in the collision?

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To find the velocity of the cars immediately after the collision, we can use the principles of conservation of momentum. The total momentum before the collision should be equal to the total momentum after the collision.

(a) Let's assume the mass of the first automobile is m1 and its velocity is v1, and the mass of the sports car is m2, and its velocity is v2.

The initial momentum of the first automobile is given by P1 = m1 * v1

The initial momentum of the sports car is given by P2 = m2 * v2

After the collision, the total momentum of the system should be conserved, so we have:

P_total = P1 + P2

Since the cars collide and lock bumpers, their velocities become the same after the collision. Let's denote the final velocity of the cars as vf.

The final momentum of the first automobile is P1f = m1 * vf

The final momentum of the sports car is P2f = m2 * vf

According to the conservation of momentum:

P_total = P1f + P2f

m1 * v1 + m2 * v2 = m1 * vf + m2 * vf

Simplifying the equation:

vf = (m1 * v1 + m2 * v2) / (m1 + m2)

Therefore, the velocity of the cars immediately after the collision is given by:

vf = (m1 * v1 + m2 * v2) / (m1 + m2)

(b) To find the percentage of the initial kinetic energy lost in the collision, we need to calculate the initial and final kinetic energies. Kinetic energy is given by the formula:

KE = (1/2) * m * v^2

The initial kinetic energy, KE_initial, is the sum of the kinetic energies of both cars:

KE_initial = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2

The final kinetic energy, KE_final, is the sum of the kinetic energies of the cars after the collision:

KE_final = (1/2) * m1 * vf^2 + (1/2) * m2 * vf^2

The percentage of kinetic energy lost, E_loss, is given by:

E_loss = ((KE_initial - KE_final) / KE_initial) * 100

Simplifying the equation:

E_loss = ((1/2) * (m1 * v1^2 + m2 * v2^2 - (m1 + m2) * vf^2)) / (1/2) * (m1 * v1^2 + m2 * v2^2) * 100

Therefore, the percentage of the initial kinetic energy lost in the collision is given by:

E_loss = ((m1 * v1^2 + m2 * v2^2 - (m1 + m2) * vf^2) / (m1 * v1^2 + m2 * v2^2)) * 100