The purpose of this experiment was to experimentally investigate the properties of projectile motion and the independence of horizontal and vertical motion. Using the knowledge of projectile motion, a prediction was made for the impact location of a freely falling ball on an incline.
Apparatus:
The apparatus used in this experiment consisted of two aluminum v-channels, one sloped and one horizontal, as shown in Figure 1 above. The use of this equipment was to give a steel sphere momentum while only giving it motion in the horizontal direction before the ball gets into free fall. Blocks and a metal stand were used to hold up the slide, and tape was used to hold the pieces together as well as to mark the origin of the steel ball's location upon sliding. A plumb bomb, which consists of a weight tied to a string, was also used to determine the exact point on the ground the steel ball begins to fall freely from the apparatus.
As shown in Figure 2, a wood plank was used as the slope required for the second part of the experiment. In order to mark the place the ball came into contact with the slope (and the floor for the first part of the experiment), a carbon copy page was used, as shown on the plank above. A meter stick was also used to measure distances in both part 1 and part 2 of the experiment. Lastly, a magnetic protractor was used to measure the angle of the inclined board.
Procedure:
Once the apparatus was set up, the steel ball was launched off from the apparatus at the marked point on the inclined v-channel, taking into consideration its impact point on the floor. Once the impact point was found, a carbon copy page was taped onto the floor at the location of the impact point. Then, the steel ball was launched off five more times at the same marked point each time, taking into account the fact that the ball should land in nearly the same place each time. The height of the apparatus and the distance the ball had traveled in the horizontal direction before the impact were measured.
The horizontal distance was measured five times, once for each trial. Then, the average distance was calculated, with the uncertainty as the range of the measured distances from the average. The average distance the ball traveled was found to be 50.50 ± 0.30 cm. The height the ball traveled from the apparatus to the floor was measured to be 94.20 ± 0.10 cm. Using these dimensions, the initial velocity in the horizontal direction was measured.
Figure 4: Dimensions measured for the free fall of the steel ball
The initial velocity in the x direction was calculated to be 115.2 cm/s. Obviously, the initial velocity in the y direction is zero.
In the second part of the experiment, an incline was used and it was predicted where the ball would land on the incline. The angle of the incline was measured to be 49º ± 2º. The distance from the end of the v-channel to the end of the board on the ground was measured to be 79.70 cm.
Using the known dimensions and components of projectile motion (acceleration in y direction, velocity in x direction, etc.), an expression for the distance d the ball will travel along the board before impact was derived. The expression is shown below:
With this expression, the theoretical distance the ball would travel relative to the board before impact was calculated and found to be 47.47 cm.
With the predicted distance in mind, the carbon copy page was pasted onto the board in the area the ball was expected to land. Then, with the apparatus set up as shown in Figure 2, the ball was launched off the v-channels in the same position as in part 1, and the experiment was run five times.Then, the distances the ball traveled along the board before impact were measured for each trial, and like in part 1, the average was calculated with the uncertainty as the range of the impact points from the average.
The average distance d the ball traveled was 48.02 ± 0.52 cm. That is a 1.15% difference from the predicted value, which proves that the prediction was very accurate and our model a good representation of projectile motion.
Discussion:
Since the equipment used in this experiment contains uncertainty, it follows that our predicted value for d also contains uncertainty. To calculate the uncertainty, the expression of d must be changed into the variables that are the source of uncertainty. In this case, those variables are "x", "y", and alpha. To get d to be a function of x, y, and alpha, the expression for Vo (shown in Figure 10 above) must be substituted into the expression for d, as shown below:
With this expression, the partial derivative with respect to x, y and alpha can be taken and the uncertainty can be solved for using the formula of "dd" shown in figure 10 above.
As shown in figure 12, the partial derivatives with respect to x, y and alpha were derived, then multiplied by dx, dy and dalpha, respectively, and summed up together.Plugging in the values for x (79.70 cm), y (94.20 cm), alpha (49º), dx (0.30 cm), dy (0.10 cm), and dalpha (0.035 rad), it was found that the uncertainty in d (dd) is:
dd = 0.564 - 0.0504 + 5.252 = 5.766
Therefore, the actual theoretical value for d is 47.47 cm ± 5.766 cm. Fortunately, the experimental value is in that range and quite close to the theoretical value for d.
Conclusion:
The sources of uncertainty in this experiment are from the equipment used for measurements. This includes the meter stick and the magnetic protractor. In addition, the range of the values in the five trials is also a source of uncertainty in the experiment. The uncertainty in d is fairly large (5.766; 12.1% of the value of d) only because of the uncertainty in alpha. As shown in the calculation for dd above, most of its uncertainty comes from the uncertainty in dalpha. This is because the uncertainty of 2º (4.1% of the value of alpha = 49º) in the measurement of alpha is very large compared to the uncertainty in the measurement of other values. This uncertainty comes from the weak accuracy in the magnetic protractor. In order to lower the uncertainty, a more accurate protractor with a much smaller uncertainty must be used. Other than the use of the inaccurate protractor, the uncertainty was small and the accuracy of the experimental to the predicted value of d was strong.
The sources of error in this experiment include the ball not being let go from the same exact position each trial, resulting in a change of the initial horizontal velocity and therefore the horizontal distance the ball travels. Another source of error is the measurement of the initial position the ball begins to free fall relative to the ground. This results in either a smaller or large horizontal distance the ball travels, changing the calculated initial velocity of the ball. Another source of error is air resistance. This error affected the experiment only in measuring the initial velocity of the ball in the horizontal direction. Since air resistance was not taken into account, the calculated velocity is probably smaller than the actual velocity since air resistance causes the projectile to travel a smaller horizontal distance. Because the calculated initial velocity was used in the second part of the experiment, it resulted in a theoretical value for d smaller than the actual, which was seen in the experiment. However, the error is not very large since the projectile has a small attacking area (area of contact with the air as it travels through the medium) and it was not in free fall for a very long time either.
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