A 821-kg car stopped at an intersection is rear-ended by a 1810-kg truck moving with a speed of 11.5 m/s. If the car was in neutral and its brakes were off, so that the collision is approximately elastic, find the final speed of both vehicles after the collision.

To find the final speed of both vehicles after the collision, we can use the principles of conservation of momentum and kinetic energy.

1. Conservation of Momentum:
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. The momentum (p) of an object is given by the product of its mass (m) and velocity (v).

Before collision:
Momentum of the car = (mass of the car) x (velocity of the car) = 821 kg x 0 m/s = 0 kg m/s
Momentum of the truck = (mass of the truck) x (velocity of the truck) = 1810 kg x 11.5 m/s = 20815 kg m/s

After collision:
Let the final speed of the car be Vc, and the final speed of the truck be Vt.
The total momentum after the collision is:
Total momentum = (mass of the car) x (final velocity of the car) + (mass of the truck) x (final velocity of the truck)

According to the conservation of momentum, the total momentum before and after the collision should be equal.
0 kg m/s + 20815 kg m/s = 821 kg x Vc + 1810 kg x Vt
20815 kg m/s = 821 kg x Vc + 1810 kg x Vt

2. Conservation of Kinetic Energy:
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

Before collision:
Kinetic energy of the car = (1/2) x (mass of the car) x (velocity of the car)^2 = (1/2) x 821 kg x 0^2 = 0 J
Kinetic energy of the truck = (1/2) x (mass of the truck) x (velocity of the truck)^2 = (1/2) x 1810 kg x (11.5 m/s)^2 = 114560.75 J

After collision:
Total kinetic energy = (1/2) x (mass of the car) x (final velocity of the car)^2 + (1/2) x (mass of the truck) x (final velocity of the truck)^2

According to the conservation of kinetic energy, the total kinetic energy before and after the collision should be equal.
0 J + 114560.75 J = (1/2) x 821 kg x Vc^2 + (1/2) x 1810 kg x Vt^2

Now, we have two equations with two variables. We can solve these equations simultaneously to find the final velocities Vc and Vt.

Solving these equations will give us the final velocities Vc and Vt, which represent the final speed of the car and truck after the collision.