15-Mar-23: Lab 5 - Trajectories

PURPOSE:  To use data from a projectile motion experiment to predict the location that a ball will land on an inclined board, and verify this mathematically.

PROCEDURE:

For this lab we set up a system so that a small ball rolls down a ramp, over the horizontal track section, and launches off the edge landing some distance away from the table on the floor.  We took note of where the ball hit the floor, then taped carbon paper on top of a blank paper so that when the ball hits, it makes a mark on the paper.  We did this 5 times, making sure the ball landed in virtually the same spot every time.

Set up to launch the ball.

5 points where ball landed.

By measuring the horizontal distance from the bottom of the table to where the ball hit the floor and the vertical distance from how high the ball was launched to the floor, we can find the launch speed of the ball.

We first used a kinematics equation for acceleration in the y direction, since initial velocity in the y direction is 0 m/s.  With this we found time, and plugged into an equation to find initial velocity in the x direction. 

Time: 0.438 s

Initial velocity in the x-direction: 1.67 m/s

Next, we imagined a situation in which an inclined board was placed along the edge of the table from where the ball was launched.  

We worked out equations to predict the distance, d, the ball would hit the sloped incline from the top of the board.

Equation for the distance along the board that the ball will hit, given v0 and α.

We then ran the experiment, again placing carbon paper over a blank paper where we expected the ball to land (based on location from previous experiment).  We used an app on our phones to measure the angle the inclined board made with the floor, which was 48°.  We measured the distance from the top of the board that the ball hit, and this experimental value was 98.5 cm.

Plugging in our initial velocity and angle, we find our theoretical value of d to be 94.5 cm.

Our theoretical value is 4% off from our actual experimental value, so we can further our experiment by taking the partial derivatives with respect to each component in our equation.  However, since x and y were directly measured by us in the experiment, we want to modify our equation to use x, y, and α.  To do this we solved for g and velocity and substituted back into our equation:

Now, we have an equation for distance but in terms of x, y, and α.  We can find the propagated error in distance by taking the partial derivative of each component and multiplying it by how uncertain we were with our measurements.  For x and y, I used a distance of 0.25 centimeters, and for α I used 1°, but converted to radians which is 0.017 rad.

The following equations show the partial derivatives with respect to each component, and then I plugged in x, y and α.  I multiplied this result by the measures of uncertainty, and added these three values to get the total propagated error.

***Each partial derivative should be in absolute values, because we want the magnitude in order to get the greatest sum.

The results show that experimental distance should fall within 1.55 cm of the theoretical distance.  Our experimental distance was 98.5.

CONCLUSION:

After calculating the propagated error, we still found that the results did not fall within our accepted error.  This could have been due to the fact that our wooden board may have moved a little, or the board was not directly lined up with the edge of where the ball was leaving the horizontal.  Our measurements may have been a little off, because it was difficult to exactly measure the horizontal and vertical distances in completely straight lines.  Something I could do to make propagated error larger is to increase the measurement uncertainty.  Instead of using 0.25 cm, I could have used 0.50 cm.

Despite some flaws, this lab helped us to see that we could use data from projectile motion to predict where a ball would hit a surface at a certain distance, and then confirm this distance mathematically.