03-Mar-2015: Trajectories

Purpose: The purpose for this lab was for us to use our understanding of projectile motion to predict the impact point of a ball on an inclined board.

Procedure: Materials required to perform this lab included an aluminum "v-channel", steel ball, board, ring stand, clamp, paper, and carbon paper. The set-up required that we make an initial incline ramp connected to a horizontal ramp on a table which shoots the steel ball off of the table onto the floor. After witnessing where the ball seems to hit the floor, we taped a sheet of paper to the floor and layered it with the carbon paper, which is used to make a mark wherever an impact occurs.

Ramp set up on table with carbon paper on floor

You can see the lab apparatus in the picture above and the close up of just the ramp below. We taped the ramps together with regular masking tape. Then we measured the angle of the incline and found it to be 31.5 degrees. We also measured the height from the end of the ramp and the floor and after 5 separate runs with the steel ball from the same starting point, we measured the average horizontal distance the ball traveled from the end of the ramp. These variables are important pieces for our calculations we do solve for the velocity of the steel ball as it exits the ramp.

closer view of ramp 31.5 degree angle 

After 5 runs you can see the 5 separate marks made onto the carbon paper by the steel ball's trajectory.

carbon paper with 5 separate ball marks

removed carbon paper

Results/Analytics: After removing the carbon paper we can find the dots which were all relatively within the same spot. We took the average distance as our value for x in our calculations for velocity. Using our known values, we were then able to create a system of equations that allowed us to solve for an initial velocity of 1.55 m/s.

Calculation for finding velocity(initial) off table

After finding the initial velocity of the ball as it exits the ramp, we had to put a board inclined at the edge of the table, such that now the ball would hit some point on the board instead of the floor. That distance from the end of the ramp to the impact on the board is our distance d. We were tasked with deriving an expression that would allow you to determine the value of d given that we knew v initial and the angle of the inclined board and the floor.

Deriving distance on board with propagated error

calculations for individual derivatives

our analytical results compared to actual measured

Our predicted value of 0.8143 plus/minus 0.0155 cm did not catch the 0.857 measured distance of the mark on the board from the end of the ramp by a few hundreths of a cm.

Conclusion/Uncertainty: Through this lab, we were able to derive the initial velocity of a steel ball as it left the end of a ramp off a table by measuring the horizontal and vertical distance it traveled and the angle of the incline that provided the steel ball with a force. After that we could calculate the exactly where the ball would land if a board was in the way of the floor as long as we were given an angle of incline on the board. Although we did not get the distance to match our range of distances, the results were not too far off and reaffirm the precision of the analytical data and its relation to the physical world.

As far as uncertainty goes, there are several factors that we do neglect like the release point of the ball not always precisely the same every time. Also keeping the carbon paper on the board was a little harder to keep stable. I dont think air resistance makes a difference in our results, it was more about understanding projectile motion in two dimensions. Of course lab equipment has certain limits which is why we rely on propagated uncertainty in order to maintain integrity in our end results.