23-March-2015: Trajectories

Purpose: We are going to utilize our understanding of projectile motion to predict where a ball rolling down a ramp will land on an inclined board.

There will be two parts to this experiment. First we will construct an apparatus that will allow us to roll a ball down a ramp and off the table.

Once our apparatus was built. We rolled the ball to see where it would land so that we may lay down carbon paper to help keep track of the landing. We rolled the ball from the same place five times down the ramp and each time the ball landed on the carbon paper it left a mark that we will use to take measurements.

The data we will use is the length in the horizontal and vertical direction from which the ball leaves the ramp. The vertical length does not change and we recorded 0.96 m. The horizontal length we gathered a range from launching the ball five times from the same place, 0.765 m to 0.785 m.

With length in both the y and x direction we used kinematic equations to calculate the time in which the ball will land on the ground when leaving the table. Using that same time we calculated the initial velocity from which the ball leaves the table.

Time it took to hit the ground 0.443 s.
The initial velocity in the x direction is 1.75 m/s.
Note that the initial velocity in the y direction is zero because at the moment the ball leaves the ramp the only momentum propelling the ball forward is all in the x direction.


Second part of this experiment is where a wooden board will be leaning at the edge of the table. Keeping in mind that the wooden board will be as close the edge of the ramp not allowing any extra length from which the ball leaves the ramp.

Launching the ball from the same spot as the first part. We will be deriving an expression that will allow us to determine the value of "d". We know our initial velocity from before and another key data to record is the angle the board is from the horizontal. Using the phone application the can measure angles we recorded our angle to be 49 degrees.

Now we derived our expression as so.

Using kinematics we will be combining the x and y and rearrange in terms of d.

Again with our initial velocity and angle we were able to calculate where the ball should land on the board. Our calculated distance was 1.0959 m.

Now we rolled the ball down the ramp onto the carbon paper tapped onto the wooden board five times. This gave us a range we measured 0.98 m  to 1.015 m.

As you can see our calculated value for the distance was slightly larger than what we measured from the carbon paper.

We will now use propagation to find if our uncertainty of distance that will help out our calculated value with the measured values. Our uncertainty for measuring the distance in the x and y direction is +/- 0.02 m. The uncertainty for the angle was +/- 0.0017 in radians.

Before we can get to propagating an uncertainty we need to create an expression that will give us values for x, y and the angle.

Once we found a suitable expression we can use. We found our uncertainty.

Conclusion

Putting it all together our calculated distance with the uncertainty was.

1.0959 m +/- 0.086 m

Now comparing it to the measured values. We calculated where our ball would land within the ranges and we were successful in meeting with the measured values.

Granted we were a bit on the higher end of the values. That could be interrupted by error from either how much gap there may have actually been between the wooden board and the ramp from which the ball leaves the table. People were walking around our table on which our apparatus was on and may have bumped it causing it to move. It is also possible that the wooden block slid.