a particle of mass m moving with a speed v meters per second strikes a particle of mass 3m at rest and joins together. express the final kinetic energy as a fraction of initial energy

To solve this problem, we can use the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.

Let's assume the initial momentum of the moving particle is p1 and the final momentum after the collision is p2. We can express the initial and final kinetic energy in terms of momentum using the following formulas:

Initial momentum (p1) = mass (m) * velocity (v)
Final momentum (p2) = total mass (m + 3m) * final velocity (v')

According to the conservation of momentum:
p1 = p2

Substituting the momentum values into the equation:
m * v = (m + 3m) * v'

Simplifying this equation:
m * v = 4m * v'

Now, let's calculate the final velocity (v') in terms of the initial velocity (v):
v' = (m * v) / (4m)

Since kinetic energy is directly proportional to the square of velocity, we can use the following formula to calculate the final kinetic energy (Kf) in terms of the initial kinetic energy (Ki):

Kf = (m * v'^2) / 2
Kf = (m * ((m * v) / (4m))^2) / 2

Simplifying the equation further:
Kf = (m * (v^2) / (16)) / 2
Kf = (m * v^2) / 32

Finally, we can express the final kinetic energy (Kf) as a fraction of initial kinetic energy (Ki) by dividing Kf by Ki:

Kf / Ki = ((m * v^2) / 32) / (m * v^2)
Kf / Ki = 1 / 32

Therefore, the final kinetic energy is 1/32 of the initial kinetic energy.