23-March-2015: Trajectories Lab

Lab 7: Trajectories

Purpose: To use understanding of projectile motion to predict the impact point of a ball on an inclined board.

Materials: Aluminum "v-channel", steel ball, board, ring stand, clamp, paper, carbon paper

Apparatus:

For this lab, we used the apparatus shown above. The channels were set up as to form an incline; so the ball would gather speed and fall to the floor. As seen below, there is a piece of paper and carbon paper taped to the floor. This was used to record the ball's impact point.

Procedure:
In order to predict the impact point of a ball on an inclined board, we first determined the ball's initial velocity.

 

Finding Initial Velocity (Part 1)

First, we measured the distance from the floor up to the v-channel. Then, we launched the ball repeatedly from the same point on the incline. The ball then traveled along the channel and fell to the floor. We repeated this step five times.

The ball landed around the 50 cm mark each time. In order to appropriately record the horizontal distance, we hung a plumb bob from the edge of the v-channel down to the floor and set that point as our 0 cm-mark. Next, we took an average of the distances between the impact points and our 0 mark.

• Measured Vertical Distance (height): 94.2 +/- .1 cm
• Average Horizontal Distance: 50.5 +/- .3 cm

Now, we determined the initial velocity of the ball with kinematics equations.

• Equations used: Height = V0y + (1/2)at^2 , Horiz. Dist. = V0x*t

First, we determined the ball's flight time. We used Height = V0y + (1/2)at^2, and solved for t.

• Height= 94.2 cm
• V0y= 0 cm/s
• a= 981 cm/s
• t= .438 sec

Then, we determined the ball's initial velocity. We used Horiz. Dist. = V0x*t, and solved for V0x.

• Horiz. Dist. = 50.5 cm
• t = .438 sec
• V0x = 115.235 cm/s or 1.15 m/s

Now, with the ball's initial velocity, we can finally predict the impact point of a ball on an inclined board.

 

Determining Impact Point (Part 2)

 

In this portion of the lab, we used the same apparatus; but we added an inclined board to the setup and measured the angle of incline. The impact point of the ball is now on the board and not on the floor. Our job is to determine this point (d).

• Measured Angle of Incline: 49 +/- 2 degrees



 

 

First, we solved the point of impact theoretically.

 

• Equations used: Height = V0y + (1/2)at^2 , Horiz. Dist. = V0x*t

We determined the ball's flight time by using Height = V0y + (1/2)at^2, and solved for t.

• Height= -d*sin(49)
• V0y= 0 cm/s
• a= -981 cm/s
• t = (d*sin(49)/490.5)^1/2

Next, we determined the point of impact using Horiz. Dist. = V0x*t and solved for d.

• Horiz. Dist.= d*cos(49)
• V0x= 115m/s or 1.15 m/s
• t = (d*sin(49)/490.5)^1/2

The calculated point of impact (d) is 47.47 cm

 

Calculated Uncertainty:

 

 

Measurements taken from previous step:

• x = 50.5 cm, dx = .30 cm
• y = 94.2 cm, dy = .10 cm
• alpha = 49 degrees, d(alpha) = 2 degrees = .035 radians



The point of impact is dependent of three variables (x ,y ,alpha). Therefore, we took the partial derivative of each variable and multiplied it by that variable's uncertainty. We, then added the products to determine the uncertainty.

• Calculated Value of Point of Impact: 47.47 +/- 5.88 cm 

 

Now, we ran an experiment to find the value of d. The procedure was the same, as in the previous part of the lab. The only difference is that point of impact is now on the board. We dropped the ball from the same point five different times and recorded the points of impact on the board. Just as we did in the previous step, we used paper and carbon paper to pin point the point of impact. We then calculated the average.

• Measured Average of Point of Impact: 48.02 +/- .52 cm

Measured vs. Calculated Point of Impact

•  Measured Average of Point of Impact: 48.02 +/- .52 cm
• Calculated Value of Point of Impact: 47.47 +/- 5.88 cm 

Our calculated point of impact falls within our measured point of impact. However, our calculated uncertainty seems a bit large for the experiment. This could be due to sources of error, such as: measured angle of incline, measured height of v-channel and measured horizontal distance ball traveled. There can also be error, if the ball was not launched from the same location on the inclined v-channel. This can also be the reason why our uncertainty is so large. Other than that, the measured and calculated point of impact are within reason.