Solving Kinetic Energy Change in Inelastic Collisions

[buffgilville]

Can someone help me with this:

For a completely inelastic collision, the fractional change in kinetic energy can be found as a function of the masses of the projectile and target carts only. Show that the fractional change in kinetic energy is given by:

[KE(final) - KE(initial)] / KE(initial) = -M / (m + M)

I know that KE = 1/2 mvsquared
but I can't seem to cancel v out to get -M / (m + M)

 

[Tide]

Don't forget that momentum is conserved in the collision!

 

[wais]

Sure, I can definitely help you with this problem! Let's start by looking at the equation for kinetic energy, which is KE = 1/2 mv^2. In an inelastic collision, the kinetic energy is not conserved and some of it is lost in the form of heat, sound, or deformation of the objects involved. This means that the final kinetic energy will be less than the initial kinetic energy.

To solve for the fractional change in kinetic energy, we need to first find the initial and final kinetic energies. The initial kinetic energy, KE(initial), is simply 1/2 mv^2, where m is the mass of the projectile and v is its initial velocity. The final kinetic energy, KE(final), can be found using the conservation of momentum equation, which states that the total momentum before the collision is equal to the total momentum after the collision. This can be expressed as:

mvi + Mvi = (m + M)vf

Where m and M are the masses of the projectile and target carts respectively, vi is the initial velocity of the projectile, and vf is the final velocity of the combined carts after the collision.

We can rearrange this equation to solve for vf, which is the final velocity of the combined carts:

vf = (mvi + Mvi) / (m + M)

Now, we can plug this value for vf into the equation for kinetic energy (KE = 1/2 mv^2) to find the final kinetic energy, KE(final):

KE(final) = 1/2 (m + M) [(mvi + Mvi) / (m + M)]^2

Simplifying this equation, we get:

KE(final) = 1/2 [(mvi + Mvi)^2 / (m + M)]

Now, we can plug in the initial kinetic energy (KE(initial) = 1/2 mv^2) into the original equation for the fractional change in kinetic energy:

[KE(final) - KE(initial)] / KE(initial) = [1/2 [(mvi + Mvi)^2 / (m + M)] - 1/2 mv^2] / 1/2 mv^2

Simplifying this equation, we get:

[KE(final) - KE(initial)] / KE(initial) = [(mvi + Mvi)^2 - mv^2] / mv^2(m + M)

Now, we can factor out