The purpose of this experiment was to find the initial velocity of a projectile in a ballistic pendulum using the conservation of momentum and energy principles.
Apparatus:
Figure 1: Ballistic pendulum used in this experiment
The apparatus of this experiment consisted of a ballistic pendulum that contains a protractor that begins at and a lever that reaches at the end of the block. It also contains a spring gun that shoots a small metal ball. When the moving ball inserts itself into the block, the block moves up along the protractor and moves the lever with it as well. When it reaches its maximum height, the lever stays at where it was pushed up along the protractor, giving the reading of the displacement of the block via an angle.
Procedure:
First, the mass of the block, the mass of the ball, and the length of the string holding the block were measured. The mass of the block was recorded as 80.9 +/- 0.1 grams. The mass of the ball was found to be 7.63 +/- 0.01 grams. Lastly, the length of the string was found to be 20.2 +/- 0.1 cm.
Next, the actual experiment was performed where the ball was shot into the block and the block moved up to a certain degree. The angle that the block went up was 17.5 +/- 0.5 degrees.
After the experimental part was performed, an expression for the initial velocity of the projectile (the metal ball) was calculated using the derivation shown below:
Figure 1: Diagram showing the before and after conditions of the ballistic pendulum
Using the conservation of momentum, the relationship between the final velocity of the block and ball and the initial velocity of the ball was found.
1.
Then, using the above relationship and the conservation of energy principle, the expression for the initial velocity of the ball was found.
2. 
The starting condition in the conservation of energy derivation is the ball implemented inside the block and it has an initial velocity Vf from the impact between the ball and block. The final condition is the ball and block reaching the highest angle along the swing before coming back down again. Plugging in the measured values for the angle, length of string , and the masses of the block and ball, the initial velocity of the ball was found.
3. 
Discussion:
The ball, right before impact, was initially travelling at 4.97 m/s. In order to determine the accuracy of this result, the propagated uncertainty in the initial velocity of the ball was found. The expression of the uncertainty in Vo (dVo) is shown below:
Figure 2: Expression of the uncertainty in the initial velocity of the ball
The expressions for the partial derivatives of Vo with respect to the length of the string, the angle, the mass of the ball and the mass of the block were found, and are shown below:
Figure 3: Expression of the partial derivative of Vo with respect to L
Figure 4: Expression of the partial derivative of Vo with respect to theta
Figure 5: Expression for the partial derivative of Vo with respect to m1
Figure 6: Expression for the partial derivative of Vo with respect to m2
Plugging in the experimental values, we obtain the following values for each partial:
Figure 7: Partial derivative values
Next, plugging these values into the expression for the uncertainty of the initial velocity of the ball, the value for the uncertainty is found:
Figure 8: Uncertainty in Vo
Therefore, the initial velocity of the ball was 4.97 +/- 0.15 m/s. The uncertainty is 3.0% of the initial velocity, proving that the value is accurate and that the use of the conservation of energy and conservation of momentum principles was an accurate method to find the initial velocity of the ball before it struck the block.
Conclusion:
The sources of error/uncertainty in this experiment were very small. The major sources of uncertainty were the uncertainty in the measurement of the length of the string attached to the block, the reading of the angle the block went up, and the measurements of the masses of the block and ball. Other sources of error that are negligible are air resistance to the ball before impact and to the black and ball after impact as well as friction when the ball went into the block.
The uncertainty in the initial velocity of the ball was very low (3.0%), proving that the experiment was successful and that the method used (conservation of energy and momentum) was an accurate method. The initial velocity of the block was 4.97 +/- 0.15 m/s.
No comments:
Post a Comment