27-April-2015: Ballistic Pendulum Activity

Lab 15: Ballistic Pendulum Lab

Purpose:  To find Vo+/-dVo of a projectile in a ballistic pendulum.

Apparatus:  The ballistic pendulum contains a protractor that begins at the end of the block. A lever will move along with the block and give us a reading of an angle once the block stops.

In the apparatus shown above, a spring gun shoots a ball into the block. The block then moves along the protractor and stops at its max height. During this process, the lever moves along with the block and stops at the max height. This angle will help us determine the displacement of the block.

Procedure:
Measured Data:

• Mass of block(m2) = 80.9+/-0.1 grams
• Mass of ball(m1) = 7.63+/-0.01 grams
• Length of String (L) = 20 +/-0.1 cm

Next, we ran the experiment to find the angle the block moved along the protractor.

• Measured Theta = 17+/-0.5 degrees

We used this reading to determine the height displacement of the block.

We used the setup shown above to determine the height displacement of the block (h).
In this case: h = L-Lcos(theta)

• Calculated height = .0087 m

The next step is to find Vo. In order to find Vo, we used conservation of momentum and energy equations.

First, we solved our conservation of momentum equation for Vf. We then plugged Vf as "v" into our conservation of energy equation and solved for Vo.

Finally, we plugged in our known values into our equation for Vo and solved for Vo.
Known values:

Mass of block(m2) = 80.9+/-0.1 grams

Mass of ball(m1) = 7.63+/-0.01 grams

Length of String (L) = 20 +/-0.1 cm

Measured Theta = 17+/-0.5 degrees

Calculated height = .0087 m

Vo = 4.80 m/s

 

Now, we must solve for the propagated uncertainty in Vo since each of our values contains some uncertainty. As you can see, Vo depends upon m2,m1,L, and theta. Therefore, Vo(L,theta, m1,m2). The equation for dVo can be seen above.

dVo is the sum of the product between the partial derivative of a given in respect to Vo multiplied by that variable's uncertainty.

The picture above shows the partial derivative for each variable in respect to Vo.

Uncertainties for each variable:

• dL = .001 m
• d(theta) = .00873 degrees
• dm1= .00001 kg
• dm2= .0001 kg

Using the known values and equations stated above, we solved the partial derivative of each variable.

• dVo/dL = 16.98
• dVo/d(theta)= 16.07
• dVo/dm1= -575.12
• dVo/dm2= 54.24

Finally, we add the products of each partial derivative multiplied by that variable's uncertainty to get dV0.

• dVo = (16.98*.001)+(16.07*.00873)+(575.12*.00001)+(54.24*.0001)
• dV0 = 0.17 m/s

Therefore, the initial velocity of the projectile was 4.80 +/- .17 m/s

 

Conclusion:

In this lab, we found the initial velocity of a projectile in a ballistic pendulum. We solved for Vo using the principles of conservation of momentum and energy. Furthermore, the sources of error and uncertainty for this lab were accounted for in finding dVo. In finding dVo, we accounted for the uncertainty in all of the measurements: mass, length, theta.  We found that the uncertainty of Vo of the projectile was 3.5%. This percentage error is very low and shows us that our value of Vo is accurate.