### Free-fallin’

**Activity: Free-Fallin’**

**Purpose: **To serve as an introduction to gravitational acceleration and the equivalence principle. Students will determine if a ball’s composition and the height at which an object is dropped affects its gravitational acceleration.

**Procedural overview:** Students will measure the masses and sizes of different balls to determine the balls’ densities. The students will drop each ball from a certain height, measure the time required for the balls to fall and calculate each ball’s gravitational acceleration. Additionally, students may drop a ball from different heights to determine if the ball’s distance from the ground affects its gravitational acceleration.

**Approximate class time: **30-60 minutes or longer, depending on the modifications made to the given procedure.

**Materials:**

- Activity guides, data tables and graphs for students
- Various round balls of different masses, sizes and densities. Each group should have at least four different balls. Try to avoid very hard, dense balls to avoid injury should a ball hit someone.
- Balances for measuring the mass in grams of each ball
- Rulers for measuring the diameter in centimeters of each ball
- Calculators
- A good location to drop balls, such as a stairwell. Higher drops (many meters) make it easier to measure the time of the fall.
- Tape measures to measure the distance in meters balls are dropped
- Stopwatches or cell phone timers that show fractions of a second, to time the fall of each ball

**Notes to the teacher: **

This activity may be used as an introduction to gravitational acceleration and the equivalence principle. The given procedure will depend on materials you collect and the physical space you have available for dropping the balls. As the instructions are written, students are asked to determine how many different types of balls they are testing and from what heights they are dropping the balls. Students will then need to determine which ball they are selecting to drop from multiple heights, and the specific heights the ball will be dropped from. If you have time, allow your students to help determine other parts of the procedure — see the “Make this activity more inquiry-based for students” suggestions below. This is a good activity for teaching students how to design an experiment.

The acceleration due to gravity on Earth’s surface is 9.8 m/sec^{2}. The equivalence principle states that objects of different densities experience the same gravitational acceleration when dropped, apart from aerodynamic drag effects. Since students will not be dropping the balls in a vacuum, aerodynamic drag will affect their results.

If students measure the time of fall very carefully (which is easier for longer distances and longer times), average several results together and use dense balls with little aerodynamic drag, they can get quite close to that result. It is recommended that students choose distances that give fall times between 1.0 second (approximately 5 meters) and 1.5 seconds (approximately 11.25 meters). Students can also try dropping balls shorter distances.

**Make this activity more inquiry-based for students:**

Choose one or more of the suggestions below to allow students to take more initiative in designing their experiment.

- Rather than giving students the accompanying data table, require them to create their own prior to executing the experiment.
- Have students create their own graphs of acceleration versus distance and acceleration versus density, or show students how to use a computer program to graph their data.
- Take out all or some of the equations given in the instructions and on the data table.
- Remove instructions No. 1–3 below, and allow students to determine how to find the density of an object. If the objects or balls that are being dropped are small enough, provide graduated cylinders to find the volume of the objects by water displacement.
- Remove the distance suggestion given in instruction No. 5 and encourage students to do a few initial tests to determine a distance.
- Remove instruction No. 10 and allow students to determine the number of trials they want to perform for each ball at each distance. Ask them to explain their thought process for the number of trials chosen.
- Remove the given procedure and the procedural overview, and give students the purpose of the experiment. Based on your students’ background knowledge, give them equations as needed. Ask students to define variables, write a hypothesis and create their own procedure.

**Procedure:**

1. Using a balance or scale, measure the mass M of a ball. Record the result in your data table, making sure to include units of measure.

2. Using a ruler, measure the diameter D [in centimeters] of the ball. Record the result in your data table, making sure to include units of measure.

3. Calculate the volume V = ( π /6)D^{3} of the ball. Record the result in your data table, making sure to include units of measure.

4. Calculate the density ρ = M/V of the ball. Record the result in your data table, making sure to include units of measure.

5. Decide how far you will let the ball fall, and use a tape measure to measure the distance d [in meters]. Record the result in your data table. It is recommended that you choose distances that give fall times between 1.0 second (approximately 5 meters) and 1.5 seconds (approximately 11.25 meters).

6. Drop the ball and use a stopwatch or cell phone timer to measure the time t [in seconds] for it to travel that distance. It is important to release the ball and not throw it, and to measure the time as accurately as possible. Record the result in your data table.

7. Calculate the average velocity during the fall, v_{avg} = d/t. Record the result in your data table, making sure to include units of measure.

8. Assuming the ball accelerates at a uniform rate, the average velocity should be half of the final velocity. Calculate the final velocity v_{final} = 2v_{avg} = 2 d/t. Record the result in your data table, making sure to include units of measure.

9. Calculate the acceleration during the fall, a = v_{final} /t = 2d/t^{2}. Record the result in your data table, making sure to include units of measure.

10. Repeat steps No. 6–9 to make a total of five measurements for the same ball and same distance. Record the results in your data table, making sure to include units of measure.

11. Find the average of your five acceleration measurements for that ball, a_{avg} = (a_{1} + a_{2} + a_{3} + a_{4} + a_{5})/5. Record the result in your data table, making sure to include units of measure.

12. Repeat steps No. 1–11 for other balls of different densities. Record the results in your data table, making sure to include units of measure.

13. Plot your data points for acceleration versus ball density on the accompanying graph. How does density affect the acceleration of a ball, and why?

*The ball’s density should not affect its measured acceleration. As previously noted, the equivalence principle predicts that objects of different densities should experience the same gravitational acceleration when dropped, apart from aerodynamic drag effects. *

14. How does aerodynamic drag affect the time it takes for an object to fall. Based on your data, which balls appear to be affected the most by aerodynamic drag? Explain how aerodynamic drag would affect the calculated acceleration for an object.

*The greater the aerodynamic drag on an object, the longer it will take for that object to fall to the ground. The calculated acceleration would be smaller when the aerodynamic drag is greater, because in order to calculate the acceleration, you are dividing 2d by t ^{2}.*

15. Repeat steps No. 1–11 using the same ball, but dropping it over four different distances. Measure the time and acceleration five times for each distance, and take the average of those five measurements. Record the results in your data table, making sure to include units of measure.

16. Plot your data points for acceleration versus distance on the accompanying graph. How does distance affect the acceleration of a ball, and why?

*Acceleration should be around 9.8 m/sec ^{2} regardless of distance. For short distances, the calculated acceleration may come out different, since it is more difficult for students to accurately measure the time of the fall when it is so short. Acceleration over longer distances may be less than 9.8 m/sec^{2} because aerodynamic drag becomes larger at higher velocities.*

17. How would you expect your experimental results to change if you conducted the experiment in a vacuum?

*Without aerodynamic drag, ball density should not affect the acceleration, and all balls should fall with an acceleration of 9.8 m/sec ^{2}.*

18. How would you expect your experimental results to change if you conducted this experiment on the moon or on another planet?

*The objects’ acceleration would be smaller on the moon because the moon’s gravity is about one-sixth that of Earth’s. Acceleration on another planet would also be different depending on the planet’s gravity. Aerodynamic drag could be different too, depending on whether the planet has an atmosphere and how thick that atmosphere is.*