Trajectories

Objective:
      Using the knowledge of projectile motion to predict the impact point of a ball on an inclined board.
Experimental equipment:
      The materials needed for this experiment is aluminum "v-channel", steel ball, board, ring stand, clamp, paper, and carbon paper.
How to set-up the apparatus: (Figure 1)
       1. Let the first aluminum "v-channel" rest on the clamp which is held static with the ring stand.
       2. Let the end of the first aluminum "v-channel" attach with the second one which rests on two small blocks.
       3. Using tape to attach connected points to make sure that everything is not moving.
       4. Launch the ball from a identifiable and repeatable point near the top of the inclined ramp. Then notice where it hits the floor.
       5. Using the carbon paper to record where the ball hits the floor, and keep launching the ball five times from the same place and verify that the ball lands in virtually the same place each time.

Figure 1: The apparatus of the experiment.

Data Collecting:

        We know that in projectile motion, two important things that needed to know are the height where the ball is launched and the range that the ball goes. Therefore, we first measure the height which is equal the distance from the edge of the table to ground. The range is determined by the distance from the table's edge it lands to the point on the carbon paper. 

        As we measured the height and the range, we got:

h = 94.7 cm ± 0.1 cm

d = 64.8 cm ± 0.2 cm

        Our objective in this part is to calculate the launch speed of the ball in order to use it to calculate the distance on the inclined board for the next part.

        Dimensions and calculations are shown in figure 2.

Figure 2: Calculating the launching speed of the ball. The result is v_ox = 14.7 cm/s.

      We then calculate the propagated uncertainty of the launching speed of the ball. The calculation is shown in figure 3. 

Figure 3: The propagated uncertainty of the launching speed of the ball.

The final result of the launching speed: v_ox = 14.7 cm/s = 0.147 m/s ± 0.037 m/s

      Since we have the value of the launching speed, we go to the next part to calculate the distance on the inclined board where the ball hits it. The inclined board is attached at the edge of the table such that the ball, as it launched at the same spot as before, will hit the board a distance d along the board.

     We place the board such that it touches the end of the lab table and the floor.  Attach a piece of carbon paper to the board such that it surrounds the region that we predicted the ball would hit. 

     The inclined board makes an angle ⦶ with the ground like shown in figure 4. Therefore we need to measure the angle between the inclined board and the floor.

Figure 4: Measuring the angle between the inclined board and the floor.

       Then we started doing the experiment by letting the ball launch and hit the inclined board five times, then measure the distance which is the experimental value of d. Figure 5 shows how we measure the distance on the inclined board after the ball hits it.

Figure 5: Measuring the distance on the inclined board after the ball hits it five times. 

The average experimental value of d = 77.5 cm

       Now we need to derive an equation to determine the theoretical value of d as we know v_ox and ⦶ from measurement. The derivation is shown in figure 6.

Figure 6: Derive an equation to calculate the distance d.

 D = (2v₀² * sin⦶) / (g * cos²⦶)

      Since we have an equation to calculate the distance, we just need to plug in the value of v_ox and ⦶ to solve. From the first part, we got the value of the launching speed of the ball is v_ox = 14.7 m/s. And in the second part, the angle we measured is: 

⦶ = 48.7° ± 0.1°

      Calculate the theoretical value of distance d on the inclined board. (Figure 7)

Figure 7: Calculating the theoretical value of the distance d on the inclined board.

      Calculate the propagated uncertainty of the distance d (Figure 8).

Figure 8: Calculating the propagated uncertainty of the distance d.

Final result (Theoretical): D = 76.1cm = 0.761m ± 0.004m

      Calculate the percent error: (Figure 9)

Figure 9: The percent error between experimental value and theoretical value of the distance d.

% error = +1.84%

Conclusion:

       In this experiment, we use the understanding of projectile motion to estimate where the ball hits the inclined board. In the first part, we use projectile motion to calculate the launching speed of the ball. The result we got is v_ox = 14.7 cm/s = 0.147 m/s ± 0.037 m/s. As we got the value for the speed, we continue part 2 to determine the distance on the inclined board where the ball hits it. While the experimental value we got is 77.5cm, the theoretical value which is from calculation is 76.1cm or 0.761m ± 0.004m. We calculated the percent error to see how far we are off from the "true" value. Our result is %error = +1.84% which is pretty good. The percent error shows that we didn't perform many errors while doing the experiment. Some of small errors may be human error in measurement or the ball is not launched at exactly repeatable point.