A 1.1 object traveling east at 21 m/s collides with a 4.5 kg object traveling west at 10 m/s. After the collision, the 1.1 kg object has a velocity 3 m/s to the west. How much kinetic energy was lost during the collision?

To solve this problem, we first need to calculate the initial kinetic energy before the collision and the final kinetic energy after the collision. The difference between these two values will give us the amount of kinetic energy lost during the collision.

Step 1: Calculate the initial kinetic energy (KE_initial)

The kinetic energy of an object is given by the formula KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.

For the 1.1 kg object traveling east at 21 m/s, the initial kinetic energy is:

KE_initial1 = 1/2 * 1.1 kg * (21 m/s)^2 = 242.55 Joules

For the 4.5 kg object traveling west at 10 m/s, the initial kinetic energy is:

KE_initial2 = 1/2 * 4.5 kg * (10 m/s)^2 = 225 Joules

So, the total initial kinetic energy is:

KE_initial = KE_initial1 + KE_initial2 = 242.55 J + 225 J = 467.55 Joules

Step 2: Calculate the final kinetic energy (KE_final)

After the collision, the 1.1 kg object has a velocity of 3 m/s to the west. We don't know the final velocity of the 4.5 kg object, but we can find it using the law of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.

The momentum of an object is given by the formula p = m * v.

The total initial momentum is:

p_initial = 1.1 kg * 21 m/s - 4.5 kg * 10 m/s = -23.1 kg*m/s

The total final momentum is:

p_final = 1.1 kg * -3 m/s + 4.5 kg * v_final = -23.1 kg*m/s

Solving for v_final gives:

v_final = (-23.1 kgm/s + 3.3 kgm/s) / 4.5 kg = -4.4 m/s

So, the final kinetic energy of the 1.1 kg object is:

KE_final1 = 1/2 * 1.1 kg * (-3 m/s)^2 = 4.95 Joules

And the final kinetic energy of the 4.5 kg object is:

KE_final2 = 1/2 * 4.5 kg * (-4.4 m/s)^2 = 43.56 Joules

So, the total final kinetic energy is:

KE_final = KE_final1 + KE_final2 = 4.95 J + 43.56 J = 48.51 Joules

Step 3: Calculate the kinetic energy lost

The kinetic energy lost during the collision is the difference between the initial and final kinetic energies:

KE_lost = KE_initial - KE_final = 467.55 J - 48.51 J = 419.04 Joules

So, 419.04 Joules of kinetic energy was lost during the collision.