Physics 12 Gizmo – Golf Range
"Hole in one!" are the words that every golfer dreams of hearing. But just getting the ball to land on the green, let alone roll into the hole, is a difficult task. Golfers must hit the ball at just the right velocity and angle so that it will travel the desired distance through the air. And that doesn't even account for the effects of wind and the spin of the ball!
Judging the precise trajectory of objects in flight, or projectiles, is an important skill in many sports—football, baseball, and basketball to name a few. Projectile motion has many military and scientific applications as well, from launching cannonballs to landing a probe on a distant planet.
Hole in One!
In this activity, you will use trial and error to judge the trajectory of a golf ball. Can you score a hole in one?
1. In the Gizmo™, notice the velocity and launch angle sliders, marked vinitial and θ. To hit the ball with the current settings, click Play ().
a. Click Reset (), and then select the Show paths and Show grid checkboxes. What was the shape of the ball's trajectory? How far did the ball travel?
b. Experiment with a variety of initial velocity (vinitial) settings until you hit the ball in the hole. If you have your speakers turned on, you will hear "Hole in one!" when this happens. Record your successful settings for vinitial and θ.
c. Click Reset. Set θ to a different angle, and then try to adjust the velocity setting to get a hole in one this way. Did it take a greater or smaller initial velocity to hit the ball the same distance this time? Record the new velocity and launch angle in your notes.
d. Click Reset. Using the dropdown menu at top right, change the Atmosphere setting to None. Click the Play button. What is the effect of no air on the flight of the ball? The frictional force between the air and the ball is called air resistance.
2. Click Reset and Clear paths. Now, you will determine the launch angle that results in the longest shot.
a. Before you begin, record your hypotheses. What launch angle will yield the longest shot with no atmosphere? What angle will yield the longest shot with air resistance?
b. Keeping the initial velocity the same, experiment with a variety of settings for θ until you are satisfied that you have found the optimal launch angle. What angle yields the longest ball flight? Is this what you predicted?
c. Click Clear paths. Change the Atmosphere setting to Air, and find the optimal value of θ in this case. Does adding air resistance change the optimal launch angle? Explain your answer.
d. Professionals can hit the ball at a velocity of around 70 m/s. About how far can these golfers drive the ball, with and without air resistance? Given there are about 1.1 yards in a meter, how far is this in yards?
e. With the atmosphere set to None, experiment with several pairs of launch angles that add up to 90° (such as 30° and 60°). What do you notice about these ball flights?
The Physics of Projectile Motion
To determine the flight path, or trajectory of a golf ball, you need to consider its initial velocity and the forces that are acting on it.
1. Click Reset and Clear paths. Check that Show paths and Show grid are still turned on, and select Show velocity vectors as well. Set vinitial to 100.0 m/s and θ to 45.0°. (To quickly set a slider to a particular value, type the value in the field to the right of the slider and hit Enter.) Change the Atmosphere setting to None.
a. Notice there are two velocity vectors, a blue one to represent velocity in the vertical direction (vy), and a red vector representing horizontal velocity (vx). Their magnitudes are listed next to the Show velocity vectors checkbox. What is the initial magnitude of each vector?
b. To calculate the horizontal and vertical components of velocity, you can use the following trigonometric formulas:
vx = vinitial • cos(θ)
vy = vinitial • sin(θ)
Use these formulas to find the values of vx and vy. Do the calculated values agree with the values given in the Gizmo?
2. Locate the zoom controls to the right of the grid, and click the - control once to zoom out. Click Play, wait about one second, and then click Pause (). The ball should still be on the upward part of its path.
a. What are the values for vx and vy now? For an object to change in velocity, it must be acted upon by a force. What force is causing vy to change? Does this force affect vx?
b. Turn on the Advanced features checkbox. What is the magnitude of gravitational acceleration shown on the g (m/s2) slider?
c. Click Play, and then Pause when the ball is at the top of its arc (roughly). How long has the ball been in the air at this point?
d. If the magnitude of gravitational acceleration (g) is 9.8 m/s2, how long would it take to decelerate from a vertical velocity of 70.71 m/s to a vertical velocity of 0 m/s? (Hint: Each second, the vertical velocity of the ball will decrease by 9.8 m/s.) Does this value agree with your observation in the previous question?
e. Given that it takes about 7 seconds for the ball to reach the highest part of its trajectory, how long will it take before it hits the ground? Write a prediction, and then test it using the Gizmo. This time is known as the hang time of the ball.
3. If physicists know the velocity of an object and how long it takes to arrive at its destination, they can calculate the distance traveled by solving the equation
d = v • t
Where d is the distance (m) v is the velocity (m/s), and t is time (s).
a. Given a horizontal velocity of 70.71 m/s and a time of 14.43 seconds, what is the distance?
b. Test your solution using the Gizmo. Did the ball travel as far as predicted?
4. Now you have a method of calculating the distance traveled for any combination of starting velocity and angle. Summarize your method by answering the following questions:
a. Given an initial velocity (vinitial) and angle (θ), how do you determine the horizontal and vertical components of motion (vx and vy)?
b. Given the vertical component of motion (vy) and the gravitational acceleration (g), how is the hang time determined?
c. Given the hang time and the horizontal velocity, how is the distance traveled determined?
5. Test your method with at least three new combinations of initial velocity (vinitial), launch angle (θ), and gravity (g). For each set of initial settings, predict the distance traveled and then test your answer using the Gizmo. Did your method work?
6. Challenge: In the first activity, you found that when the initial velocity is held constant, complementary launch angles (such as 30° and 60°) resulted in the same horizontal distance. Try to prove this algebraically using the equations you have developed for finding distance.
Extension Activity: Hitting off a Cliff
A classic problem in projectile motion is the trajectory of a ball launched horizontally from a cliff.
1. Click Reset and Clear paths. Check that the selected atmosphere is None. With the Advanced features checkbox turned on, set the height of the person (hperson) to 200.0 m. Set vinitial to 50.0 m/s, θ to 0.0°, and g to 9.8 m/s2.
a. Click Play. How long did it take for the ball to drop to the ground? About how far did it travel horizontally?
b. Set vinitial to 10.0 m/s, and click Play again. How long did it take the ball to fall to the ground this time? How far did it travel?
c. Does changing the horizontal velocity affect how long it takes to fall? Try several other vinitial settings to confirm your answer.
2. For an object in freefall, the relationship between the height (h), gravitational acceleration (g), and time (t) is shown by the equation
a. Use the equation to solve for t when h is 200 m and g is 9.8 m/s2. Does this agree with the experimental value?
b. Given the time spent falling and a horizontal velocity of 50 m/s, how far will the ball travel? Does this agree with the experimental value?
c. Describe a method or write an expression for determining the hang time and distance traveled for an object shot horizontally off of a cliff. Test your method using the Gizmo.
3. Challenge: If time permits, use the Gizmo to explore these other concepts:
a. How do you determine the horizontal distance a ball will travel when launched at an angle off a cliff?
b. How does air resistance affect the velocity, hang time, and horizontal distance a ball will travel?