Ballistic Spring System Problem

[Branson Holt]

Homework Statement

A 10.1 g bullet is fired horizontally into a
44.1 g wooden block that is initially at rest on
a rough horizontal surface and connected to a
massless spring of constant 97.8 N/m.
If the bullet-block system compresses the
spring by 0.441 m, what was the speed of
the bullet just as it enters the block? The
acceleration of gravity is 9.8 m/s
2
. Assume
that the coefficient of kinetic friction between
the block surface is 0.554.
Answer in units of m/s.

Homework Equations

The equation our teacher gave us to solve it is: 1/2 mv^2= 1/2kx^2 + Fd
but I've used the equation and it hasn't worked for me.[/B]

The Attempt at a Solution

Using the equation above. Is the equation right?[/B]

 

[Orodruin]

This depends on how you are using it. Can you show your work please?

 

[Branson Holt]

Well to start off I plugged in everything that I had from the information given:
1/2(.0101)v^2= 1/2(97.8)(.441)^2 + F(.441)

Then I found F by using F= u(coefficient of friction) x Fn(Normal Force)
so F= (.554)Fn which I found normal force by using Fn= (m+M)g
Fn= (.0101 + .0441)(9.8)= .53116
So when you plug that back in you get
F=(.554)(.53116)
1/2(.0101)v^2= 1/2(97.8)(.441)^2 + F(.441)
After i work this equation out solving for v I always get a consistent answer but it's not right.

 

[haruspex]

I plugged in everything that I had from the information given:
1/2(.0101)v^2= 1/2(97.8)(.441)^2 + F(.441)

There's a step before this.
Consider the process in two stages:
- merging of bullet and block
- subsequent movement of combined bullet and block
Both stages involve loss of mechanical energy, the first in impact, the second in friction.
So the first thing to find out is the speed with which the combined bullet and block start moving.

 

[HalfLostAndSearching]

Hello,

Thank you for providing the problem statement and your attempt at a solution. First, I would like to clarify that the equation provided by your teacher is correct. It is known as the conservation of energy equation and is commonly used in solving problems involving spring systems.

Now, let's break down the problem and see how we can apply the equation. We have a bullet with a mass of 10.1 g and a wooden block with a mass of 44.1 g. The system is initially at rest and the bullet is fired horizontally into the block. This means that the initial velocity of the bullet is unknown, but it can be assumed that it is greater than 0 since it is fired.

The bullet and the block compress the spring by 0.441 m, which means that the spring has done work on the system. This work can be represented as 1/2kx^2, where k is the spring constant and x is the compression distance. We can also consider the work done by the friction force, which can be represented as Fd, where F is the friction force and d is the displacement of the block due to friction.

Using the conservation of energy equation, we can set the initial kinetic energy of the system (bullet) to be equal to the final potential energy of the system (compressed spring) plus the work done by the friction force:

1/2 mv^2 = 1/2kx^2 + Fd

Substituting the given values:

1/2 (0.0101 kg) v^2 = 1/2 (97.8 N/m) (0.441 m)^2 + (0.554)(0.0441 kg)(9.8 m/s^2)(0.441 m)

Simplifying and solving for v, we get:

v = 0.463 m/s

Therefore, the initial velocity of the bullet as it enters the block is approximately 0.463 m/s.

I hope this helps. Keep practicing and don't hesitate to ask for clarification if needed. Good luck!