Ballistic Pendulum

Introduction:

In this lab we had to use the ballistic pendulum to determine the initial velocity of a projectile using conservation of momentum and conservation of energy.

To study this we use  the equation KE=PE. Kinetic energy equals the Potential Energy so energy is conserved.

 KE=PE

1/2 (M + m)(V^2) = (M+m)(g)(h)

To solve for velocity of the pendulum arm, we rearrange the equation to

V=(2gh)^(1/2)

We know that momentum is conserved so 

o

P(initial) = P(final)

or 

(M+m)((2gh)^(1/2))=mvo

Where v₀ is the initial velocity of the ball before impact. By using equations and we can find the inital velocity, of the ball.

Vo=[(M+m)(2gh^(1/2))]/m

This combines both the conservation of momentum equation and the conservation of energy equation and shows how we can find v0, which is the initial velocity of the ball 

We could also find the initial velocity of the ball if we just shoot it into the open from a known height.

so we use Kinematics to find the initial velocity 

X=Vox*t+(1/2)ax*t^2 =Vox*t+(1/2)0*t^2 = Vo*t

Y= Voy*t+(1/2)ay*t^2 = 0*t+(1/2)(-9.8m/s^2) = -4.9t^2

t = [Y/(-4.9)]^(1/2)

X=Vo*t

X=Vox*[Y/(-4.9)]^(1/2)

Vox=X/([Y/(-4.9)]^(1/2)

Data

We ran 9 trials for the ball on the pendulum and obtained the following heights in notches. Our average was 8.56 notches.

Trials	Distance (notches)
1	10
2	7
3	8
4	11
5	10
6	9
7	8
8	8
9	6
Average	8.56

Then to find the initial velocity we use

V=(2gh)^(1/2)
V=(2(-9.8m/s^2)(0.096))^(1/2)
V=1.37m/s

Vo=[(M+m)(2gh^(1/2))]/m

Vo=[(0.1994m+0.0573m)(1.37m/s)]/)0.0573m

Vo=6.137m/s

Then we shot the ball out into the open and recorded the distances it traveled

Trials	Distances X(m)
1	2.615
2	2.629
3	2.675
4	2.722
5	2.747
6	2.777
7	2.786
8	2.818
9	2.818
Average	2.7319

To calculate this we used 

Vox=X/([Y/(-4.9)]^(1/2)

Vox=2.73m/([1.006/(-4.9)]^(1/2)

Vox= 6.031

To find the percent error between the two values we did the following

{(V1-V2)/[(V1+V2)/2]}*100

{(6.031-6.137)/[(6.031+6.137)/2]}*100

(0.01742*100)

1.742% error

Conclusion

In conclusion, I learned that you can find the initial velocity of an object by combining the laws of conservation of momentum and the laws of conservation of energy. Energy is never lost , it just gets transformed into a different type of energy. We could find the initial velocity from the kinematics equations by rearranging them. We also found the initial velocity with energy equations. Our percent difference was 1.742% which is relatively low. Some sources of error could be measurement. We measured distances by stacking meter sticks. The friction of the notches was not taken into account and could have lowered the pendulums maximum height. Another friction that was not taken into account was the the part where the pendulum was attached to the stand. This was assumed to be frictionless.