PEP 294: Web-Based Labs

Lab #6: Projectile Motion

Before you start this lab, first read the general instructions for the PEP 294 web-based labs.

Procedures

A projectile is any object projected into the air. A soccer ball kicked is a projectile. The body of a long jumper in the air is also a projectile. Projectile motion is special because it is governed by the constant gravitational acceleration. Moreover, the gravitational acceleration affects only the vertical component of the motion, not the horizontal motion. It is why the trajectory of the projectile is parabolic if we ignore the effects of the air friction. One important factor which affects the trajectory of the projectile is air friction or air resistance. We ignore the effects of air resistance for simplicity, but not because it does not affect the trajectory of the projectile at all. The purpose of this lab is to understand the nature of the projectile motion (the role of the gravitational acceleration).

Step 1: Strategies

In analyzing the projectile motion, the following strategies are commonly employed:

(a) Resolve the initial velocity vector into the horizontal and vertical components.

(b) Treat the horizontal motion of the projectile separately from the vertical motion.

(c) Use the equations of projectile motion.

A group of equations called equations of projectile motion will be provided later in this lab.

Step 2: Resolution of the Projection Velocity Vector

Two initial conditions in the projectile motion are the projection velocity (magnitude + direction) and the projection position. These factors determine the trajectory of the projectile. For simplicity, you will only deal with simple projections in this lab, in which the projection height and the landing height are the same. In other words, the projection position is not important any more. Figure 1 below shows the projection velocity and the resulting trajectory of the projectile. Angle q shown in Figure 1 is the projection angle (direction of the projection velocity vector).

    Figure 1

When the projection velocity is given, one can resolve this vector into the horizontal and vertical components. You've practiced this in Lab #3 already. Let v_x and v_y be the horizontal and vertical components of the projection velocity. Then:

   Equations 1a & b

[Example 1]     A shot-putter throws a shot. The velocity of the shot is 10 m/s and the projection angle is 40 degrees. What are the horizontal and vertical projection velocities?

Using Equations 1a & b:

Therefore, the horizontal and vertical projection velocities are 7.66 m/s and 6.43 m/s, respectively.

Note that we use symbols v_x and v_y for the initial horizontal and vertical projection velocities.

Step 3: Equations of Projectile Motion

From the equations of constant acceleration we can derive the following set of equations:

   Equations 2a-e

where, t = time in flight after projection, (v_x, v_y) = projection velocity, (v_h , v_v) = velocity of the projectile at time t, (d_h , d_v) = position of the projectile at time t, g = constant gravitational acceleration (9.81 m/s²).

[Example 2]     From Example 1, find the locations and velocities of the shot at given instances and fill out Table 1.

- The initial horizontal and vertical velocities are 7.66 m/s and 6.43 m/s. Use Equations 2a, b, c & d to obtain the position and velocity of the projectile at 0.1 s after projection:

- At 0.2 s after projection:

Click here to download the Excel file for this example.

Table 1. Positions and Velocities of the Shot

    Figure 2

    Figure 3

Due to the gravitational acceleration, the vertical velocity decreases as time passes. Note that a positive vertical velocity means upward motion while a negative vertical velocity means downward motion. In other words, the shot starts moving down shortly after 0.6 s in flight. The vertical velocity at the apex is 0 m/s.

Step 4: Height, Distance & Flight Time

Figure 4 below shows the height of the apex (H), the max distance (D) and the ascending time (T) of the projectile motion. The ascending time is the time elapsed to the apex.

   Figure 4

The height of the apex and the ascending time are computed based on the fact that the vertical velocity of the projectile at the apex is 0. Since the projectile is at the apex at time T after projection, replace v_v, t, d_v in Equations 2c & e with 0, T, H:

    Equation 3

and,

   Equation 4

Note in Equations 3 & 4 that T and H are function of only the initial vertical velocity (v_y) since the gravitational acceleration (g) is constant. In other words, the initial vertical projection velocity solely determines the height and ascending time of the projectile motion, nothing else.

The trajectory of the projectile in Figure 2 has a symmetric shape, and the ascending time is the same to the descending time. Thus, the overall flight time (FT) is:

   Equation 5

The max. horizontal distance (D) is computed from Equation 2b. Since the projectile is at the landing point at time 2T after projection, replace d_h and t in Equation 2b with D and 2T:

   Equation 6

Using Equations 3 to 6, you can easily compute the height of the apex, the ascending time (or descending time), the flight time and the max. distance of the projectile motion.

[Example 3]   From Example 1, compute the height of the apex with respect to the the projection point, the ascending time and the horizontal distance the shot travels until it returns to the height of projection.

- The initial horizontal and vertical velocities are 7.66 m/s and 6.43 m/s, respectively. Using Equations 3, 4 & 6:

- Compare these with the results of Example 2.

Summary

Due to gravity, any object projected into the air falls on to the ground showing a parabolic trajectory. This type of motion is called the projectile motion. The following are the properties of the projectile motion:

- In analyzing the projectile motion, we always treat the vertical motion separately from the horizontal motion. One can easily resolve the projection velocity vector into the horizontal and vertical components using the trigonometric functions.

- The gravity only affects the vertical motion of the projectile. It is why we observe a parabolic path of the projectile.

- During the projectile motion, the horizontal velocity of the projectile does not change since there is no acceleration along the horizontal axis. The vertical velocity is altered by the gravitational acceleration (g = 9.81 m/s², downward).

- The direction of the vertical motion changes at the apex of the trajectory from upward to downward. The vertical velocity at the apex is always 0.

- Regardless of the shape of the projectile, the center of mass of the projectile always follows a parabolic path determined by the projection velocity. Once you are in the air, relative body motions using the arms and legs can not alter the trajectory of your body center of mass. (This rule equally applys to Michael 'Air' Jordan.)

- The height of the apex and the ascending time are functions of only the initial vertical velocity. In order to increase the jumping height or the flight time, one should increase the vertical component of the projection velocity.

Questions (25 pts)

A Denver Bronco attempts a field goal against Cleveland Browns with the ball placed on the tee at a distance of 29 m from the goal post. The initial velocity of the ball is 21 m/s and the kicking angle is 30°. Click here to download the Excel worksheet file. Do all the computations and graphing using Excel. Attach the spreadsheet printouts to the report.

1. Compute the initial horizontal and vertical velocities of the ball. (3 pts)

2. Compute d_h, d_v, v_h, and v_v for t = 0, 0.1, 0.2, 0.3, ...., 2.2 s and draw the trajectory of the ball. (Use the downloaded worksheet for the computations and graphing.) Based on the computations and graphing, briefly explain how the gravity affects the motion of the ball in the air? (10 points)

3. What happens to the horizontal velocity of the ball in the air and why? (2 points)

4. Draw the vertical velocity-time curve of the ball and briefly explain what happens to it and why. (5 points)

5. Compute H, T, & D. Which factor solely determines H & T of a projectile? (5 pts)

6. (Bonus Question) Will the ball clear the goal of 3 m high? Show the steps. (Hint: find the vertical position of the ball at the goal and compare it with the height of the goal.)