A 3 kg toy car with a speed of 10 m/s collides head-on with a 2 kg car traveling in the opposite direction with a speed of 2 m/s. If the cars are locked together after the collision with a speed of 5.20 m/s, how much kinetic energy is lost?

To find the amount of kinetic energy lost in the collision, we need to compare the initial kinetic energy of the two cars before the collision to the final kinetic energy of the locked-together cars after the collision.

The formula to calculate kinetic energy (KE) is:

KE = 1/2 * mass * velocity^2

Initially, we have the following information:

Mass of first car (m1) = 3 kg
Initial velocity of first car (v1) = 10 m/s

Mass of second car (m2) = 2 kg
Initial velocity of second car (v2) = -2 m/s (since it is traveling in the opposite direction)

We can calculate the initial kinetic energy of each car using the formula:

KE1 = 1/2 * m1 * v1^2
KE2 = 1/2 * m2 * v2^2

Next, we need to calculate the final kinetic energy of the locked-together cars.

Mass of locked-together cars = m1 + m2 = 3 kg + 2 kg = 5 kg
Final velocity of locked-together cars (v_f) = 5.20 m/s

We can calculate the final kinetic energy using the same formula:

KE_f = 1/2 * mass * v_f^2

To find the amount of kinetic energy lost, we subtract the final kinetic energy from the sum of the initial kinetic energy of both cars:

KE_lost = (KE1 + KE2) - KE_f

Let's calculate the values step by step:

1. Calculate the initial kinetic energy of the first car (KE1):
KE1 = 1/2 * m1 * v1^2
KE1 = 1/2 * 3 kg * (10 m/s)^2
KE1 = 1/2 * 3 kg * 100 m^2/s^2
KE1 = 150 J

2. Calculate the initial kinetic energy of the second car (KE2):
KE2 = 1/2 * m2 * v2^2
KE2 = 1/2 * 2 kg * (-2 m/s)^2
KE2 = 1/2 * 2 kg * 4 m^2/s^2
KE2 = 4 J

3. Calculate the final kinetic energy of the locked-together cars (KE_f):
KE_f = 1/2 * mass * v_f^2
KE_f = 1/2 * 5 kg * (5.20 m/s)^2
KE_f = 1/2 * 5 kg * 27.04 m^2/s^2
KE_f = 67.6 J

4. Calculate the amount of kinetic energy lost (KE_lost):
KE_lost = (KE1 + KE2) - KE_f
KE_lost = (150 J + 4 J) - 67.6 J
KE_lost = 84 J

Therefore, the amount of kinetic energy lost in the collision is 84 Joules.

To find the amount of kinetic energy lost in the collision, we need to calculate the initial kinetic energy of the system before the collision and compare it to the kinetic energy after the collision.

1. Calculate the initial kinetic energy of the system:
- The kinetic energy of an object can be calculated using the formula: KE = (1/2) * mass * velocity^2.
- For the first car, with a mass of 3 kg and a speed of 10 m/s, the kinetic energy is KE1 = (1/2) * 3 kg * (10 m/s)^2.
- For the second car, with a mass of 2 kg and a speed of 2 m/s, the kinetic energy is KE2 = (1/2) * 2 kg * (2 m/s)^2.
- Therefore, the total initial kinetic energy of the system is KE_initial = KE1 + KE2.

2. Calculate the final kinetic energy of the system:
- After the collision, the cars are locked together and have a combined mass of 3 kg + 2 kg = 5 kg, and a combined final speed of 5.20 m/s.
- The final kinetic energy can be calculated using the formula: KE_final = (1/2) * total mass * final velocity^2.
- Therefore, the total final kinetic energy of the system is KE_final = (1/2) * 5 kg * (5.20 m/s)^2.

3. Calculate the kinetic energy lost:
- The kinetic energy lost in the collision is the difference between the initial kinetic energy and the final kinetic energy: KE_lost = KE_initial - KE_final.

Now, let's plug in the values and calculate the kinetic energy lost:

KE_initial = (1/2) * 3 kg * (10 m/s)^2 + (1/2) * 2 kg * (2 m/s)^2
KE_initial = 150 J + 4 J
KE_initial = 154 J

KE_final = (1/2) * 5 kg * (5.20 m/s)^2
KE_final = 67 J

KE_lost = KE_initial - KE_final
KE_lost = 154 J - 67 J
KE_lost = 87 J

Therefore, the amount of kinetic energy lost in the collision is 87 Joules.

figure final velocity from conservation of momentum. Then, use that velocity to calcualte the final KE. Compare it with the initial KE