How Do You Calculate the Initial Velocity of a Bullet in an Elastic Collision?

[spectravoid]

Homework Statement

A bullet of mass m is shot into a block of mass M that rests on a rough surface with coefficient of kinetic friction k. The block slides for a distance of x meters before coming to a stop. Derive an expression for the initial velocity of the bullet.

Homework Equations

0.5mvi^2 - k(-1)x = 0.5(M+m)vf^2 (work energy theorem just with the variables replaced with the ones in the question)
and also this but I'm not sure where to use it Ff = (M + m)gk

The Attempt at a Solution

vi = +/- sqrt((vf^2 + (Mvf^2)/m + (2kx)/m))

I need to remove vf

thank you for help

 

[Dick]

You don't want to use the work/energy stuff here. This is an INelastic collision. When the bullet collides with the block it is stopped by friction within the block and you have no data to calculate that directly. What you can count on is conservation of momentum. The initial momentum of the bullet is equal to the initial momentum of the bullet plus block combination. This will let you get an initial velocity for the block. Go from there.

 

[m2003]

I would approach this problem by first reviewing the given information and equations. The work-energy theorem states that the initial kinetic energy of the bullet (0.5mvi^2) must be equal to the final kinetic energy of the bullet and block (0.5(M+m)vf^2) minus the work done by friction (k(-1)x). We can also use the equation Ff = (M + m)gk to calculate the force of friction acting on the block.

To derive an expression for the initial velocity of the bullet, we can rearrange the work-energy theorem equation to solve for vi:

0.5mvi^2 = 0.5(M+m)vf^2 - k(-1)x

We can then substitute the equation for force of friction into the work-energy theorem:

0.5mvi^2 = 0.5(M+m)vf^2 - (M + m)gk(-1)x

Next, we can substitute the equation for final velocity (vf) from the first equation into the second equation:

0.5mvi^2 = 0.5(M+m)((0.5mvi^2 + (Mvi^2)/m + (2kx)/m)) - (M + m)gk(-1)x

Simplifying and solving for vi, we get the following expression for the initial velocity of the bullet:

vi = +/- sqrt((vf^2 + (Mvf^2)/m + (2kx)/m))

This expression gives us the initial velocity of the bullet based on the final velocity, mass of the block and bullet, coefficient of kinetic friction, and distance traveled by the block. We can use this equation to calculate the initial velocity for different scenarios and analyze the effects of changing variables such as mass, friction, and distance.