Purpose: To demonstrate that a projectile follows a parabolic path.

Discussion: The motion of a projectile can be summarized by stating that the horizontal and vertical motions are independent of each other. The horizontal motion corresponds to motion at constant velocity, while the vertical motion corresponds to uniformly accelerated motion. If we assume that the projectile is launched horizontally (v_y0 = 0), and x₀ = 0, the x and y coordinates are given by,

x = v_x0t and y = y₀ - gt².

Solving for t in terms of x in the first equation, and substituting into the second equation,

y = y₀ - gx²/(2v²_ox)

Viewed as a function of x, y corresponds to a parabola with a vertex at the point (0, y₀) and which decreases for x > 0. Note that when y = 0, x = v_x0[2y_o/g]^1/2, which is known as the range R. If we know R, y₀, and assume g = 9.8 m/s² , we can solve for v_x0.

Procedure:

1. Measure y₀ from the floor to the bottom surface of the ball when it is on the gun. Mark the position of the table leg, and the horizontal position of the front end of the gun on the ground with pieces of tape. Attach a strip of red recording paper along the length of the target.

2. Position the target at a particular distance x from the front end of the gun. Make sure that the target is in the expected path of the metal ball. Carefully load the metal ball into the spring loaded gun. The ball is fired at a reasonably high velocity, make sure no one is in the expected path of the ball before firing! Fire at the target. Mark the point of impact of the ball on the target and measure y. Be prepared to repeat this several times until you obtain consistent results. Record x and y in the table.

3. Repeat step 2 for various values of x.

4. Make a graph of y vs. x. Make sure to clearly mark and circle the data points.

5. Using the measured range R, y₀ and g, calculate v_x0.

6. Use v_ox, y₀, and g, to calculate the theoretical value of y corresponding to the values of x which were used in the experiment. Record these in the data table.

7. Mark the theoretical values on the graph as small dots. Draw a smooth parabolic curve through these points. Unlike your experimentally measured values, these theoretical values are part of a theoretical curve, therefore it is best if they are not visible when you draw in the curve.

8. Since we assumed g = 9.8 m/s², we do not really have a particular final result which we can compare to a theoretical prediction. However, we can decide qualitatively whether or not the ball follows a parabolic path. Comparing the data points to the theoretical curve, does the projectile follow a parabolic path?