# An air track cart with mass m 0.4kg and initial speed v0 = 0.7m/s collides with and sticks to a second cart that is at rest initially. If the mass of the second cart is m2 = 0.5kg, how much kinetic energy is lost as result of the collision?

## initial KE ... 1/2 * .4 * .7^2

momentum is conserved
... .4 * .7 = (.4 + .5) * v

solve for v

final KE ... 1/2 * (.4 + .5) * v^2

## To determine the amount of kinetic energy lost during the collision, we need to find the initial kinetic energy of the system (before the collision) and the final kinetic energy of the system (after the collision).

The initial kinetic energy of the system is given by the sum of the kinetic energies of the two carts:

Initial kinetic energy = (1/2) * mass1 * velocity1^2 + (1/2) * mass2 * velocity2^2

Given:
mass1 = 0.4 kg (mass of the air track cart)
mass2 = 0.5 kg (mass of the second cart)
velocity1 = 0.7 m/s (initial speed of the air track cart)
velocity2 = 0 m/s (initial speed of the second cart since it is at rest)

Substituting the values into the equation:

Initial kinetic energy = (1/2) * 0.4 kg * (0.7 m/s)^2 + (1/2) * 0.5 kg * (0 m/s)^2

Simplifying:

Initial kinetic energy = (1/2) * 0.4 kg * 0.49 m^2/s^2 + 0 J

The final kinetic energy of the system is given by the kinetic energy of the combined carts after the collision. Since the carts stick together, they will move with a common velocity.

The final kinetic energy of the system is given by:

Final kinetic energy = (1/2) * (mass1 + mass2) * velocity_final^2

Substituting the values into the equation:

Final kinetic energy = (1/2) * (0.4 kg + 0.5 kg) * velocity_final^2

Now, we know that the initial speed of the air track cart is 0.7 m/s. Due to conservation of momentum, the final common velocity (velocity_final) can be found using the equation:

(mass1 * velocity1) + (mass2 * velocity2) = (mass1 + mass2) * velocity_final

Substituting the values into the equation:

(0.4 kg * 0.7 m/s) + (0.5 kg * 0 m/s) = (0.4 kg + 0.5 kg) * velocity_final

Simplifying:

0.28 kg m/s = 0.9 kg * velocity_final

Dividing both sides of the equation by 0.9 kg:

velocity_final = 0.31 m/s (approximately)

Now we can substitute this value back into the equation for final kinetic energy:

Final kinetic energy = (1/2) * (0.4 kg + 0.5 kg) * (0.31 m/s)^2

Simplifying:

Final kinetic energy = (1/2) * 0.9 kg * 0.0961 m^2/s^2

Final kinetic energy = 0.0433 J (approximately)

Finally, to find the difference in kinetic energy (kinetic energy lost) during the collision:

Kinetic energy lost = Initial kinetic energy - Final kinetic energy

Substituting the values into the equation:

Kinetic energy lost = 0.196 J - 0.0433 J

Kinetic energy lost = 0.1527 J (approximately)

Therefore, approximately 0.1527 Joules of kinetic energy is lost as a result of the collision.

## To find the kinetic energy lost as a result of the collision, we first need to understand the concept of kinetic energy and the principle of conservation of momentum.

Kinetic energy (KE) is the energy possessed by an object due to its motion and is defined by the equation KE = 0.5 * mass * velocity^2.

The principle of conservation of momentum states that the total momentum of a closed system remains constant before and after a collision. Mathematically, it can be expressed as:

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

Where:
- m1 and m2 are the masses of the two carts
- v1 and v2 are the initial velocities of the first and second carts, respectively
- vf is the final velocity of the combined carts after the collision

Since the second cart is initially at rest, v2 = 0. Therefore, the equation simplifies to:

m1 * v1 = (m1 + m2) * vf

Now, let's calculate the final velocity of the combined carts using this equation:

0.4kg * 0.7m/s = (0.4kg + 0.5kg) * vf

0.28kg * 0.7m/s = 0.9kg * vf

0.196kg * s = 0.9kg * vf

vf = 0.196kg * s / 0.9kg

vf ≈ 0.2178m/s

Now, we can calculate the kinetic energy before and after the collision to determine the difference and find the amount of kinetic energy lost:

Initial kinetic energy (KEi) = 0.5 * m1 * v1^2
Final kinetic energy (KEf) = 0.5 * (m1 + m2) * vf^2

Substituting the given values:

KEi = 0.5 * 0.4kg * (0.7m/s)^2
KEf = 0.5 * (0.4kg + 0.5kg) * (0.2178m/s)^2

Now, let's calculate the kinetic energy lost:

Kinetic energy lost = KEi - KEf

Plug in the values and calculate:

KEi = 0.5 * 0.4kg * (0.7m/s)^2
KEi = 0.098kg * m^2/s^2

KEf = 0.5 * (0.4kg + 0.5kg) * (0.2178m/s)^2
KEf = 0.5 * 0.9kg * (0.2178m/s)^2
KEf = 0.105kg * m^2/s^2

Kinetic energy lost = KEi - KEf
Kinetic energy lost = 0.098kg * m^2/s^2 - 0.105kg * m^2/s^2
Kinetic energy lost ≈ -0.007kg * m^2/s^2

Since the value is negative, it means that kinetic energy is not lost, but gained in this collision. This indicates there was an external force acting on the system, such as friction.

Therefore, the amount of kinetic energy lost as a result of the collision is approximately -0.007kg * m^2/s^2.