Lab 5 Trajectories

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

 
 

The apparatus is used to examine the trajectory of a metal ball and the carbon paper is used to mark were the metal ball lands on the white paper.




















Procedure:

1. Set up the ramp system like in the picture above.
2. Mark the point from which the ball will be launched on the ramp with some tape so that the ball will be released from the same point for each trial.
3. launch the ball and check to see where it lands so that you can tape the blank piece of paper with the carbon paper on top in the same spot where the ball landed.
4. launch the ball 5 more times from the same place as was marked on the ramp and verify that it lands in the same place each time.
5. Measure the height of the bottom of the ball when it launches off the ramp. And measure the distance from the table's edge that it lands.
6. From your measurements, determine the launch speed of the ball .

7. Attaching an inclined board, like the one pictured on the image to the right the ball will now hit the board a distance d when it is launched. Derive an expression that will allow you to determine the distance of d given that you have the initial velocity V0 and the angle .

8. Do a trial run to see where the ball is going to land on the board. With tape attach the white paper and carbon paper on the spot where the metal ball landed on the board. Measure the angle of elevation of the board and perform five more trials from the same spot.

9. Determine the experimental value of d and report your experimental value with uncertainty

10. Compare experimental and theoretical values for d. comment on sources of uncertainty or error in the experiment.

 

 

 

We measured the distance from the edge of the ramp to where the ball landed in the x direction. And we measured the height of the edge of the ramp to the ground in the y direction. We got the measurements to be :

 



 

distance in y direction:     .94 meters

 

distance in x direction:     .68 meters

 

Using the distances measured we were able to calculate the initial velocity of the ball when it launches by using the method to solve a projectile motion.

 



 



Splitting the projection of the ball into an x and y direction we used the kinematics equation in the image above to solve for time of flight in the y direction. Solving for the time we were able to solve for the initial velocity in the x direction and we calculated the initial velocity to be 1.55 m/s.

 

For the second part of the experiment we attached a ramp to the table and find the distance d where the ball is going to land. We derived two equations to solve for the distance d:

 

 



We calculated the initial velocity in the first part of the lab and we measured the angle of the board using an app that measures angles on our phone. So we have two equations and two unknowns the unknowns being d and t. Since the time t is the same for both x and y we solved each equation for t and made the equations equal to each other then we solved for d and got an expression that solves for d.

 

 

Calculated initial velocity :  1.55 m/s

 

Measured angle of board : 48 degrees

 

g = 9.8 m/s^2

 

 

The distance we calculated where the ball is supposed to land on the board is .814 meters. So we launched the ball 5 times and got a range between .841m and .869m.

To see if our calculated value is acceptable we have to use the method of propagated error.

 

 





To be able to calculate the propagated error we are going to have to derive an expression that involves x and y and d together. So we used the kinematics equations for x and y and we solved for t again and made the equations equal to each other. Then we solved for initial velocity so that we could plug it into our equation to get a new equation that has d, x and y.

 

 







Now that we have the appropriate expression we took the partial derivatives and calculated the range of our uncertainty in the calculation.

 

 

 

the uncertainty that we calculated was  +/- .0155 .

 

As you can see our calculated value for the distance with the uncertainty is not within the range of the measured values.

 

Calculated d and uncertainty : .8143 +/- .0155

 

Measured range: .841m - .869m

 

The reason that the calculated value might not be in the range of the measured value was that we could have been measuring the distance with the meter stick from the wrong point. There was space between the edge of the ramp and the edge of the ramp . Another factor that could have caused error was that the board was not taped to the floor and table so it could have moved and the board was also not flat the wood was bent a little.