Kinetic Energy Ball Drop

Introduction

The momentum of an object is proportional to the speed of that object. However, the kinetic energy of an object is proportional to the square of its speed. Demonstrate this principle by dropping a ball bearing into soft clay.

Concepts

• Kinetic energy
• Potential energy
• Inelastic collisions

Background

Albert Einstein’s (1879–1955) most famous equation, and possibly the world’s most familiar equation, is E = mc². What basis did Einstein have to use the c² conversion factor? The roots of this remarkably simple-looking equation go back to an 18th-century Frenchwoman. Emilie du Châtelet (1706–1749) was a mathematical genius who not only completely translated Isaac Newton’s Principia from its original Latin into French (the first person to do so), but she also resolved a long-standing debate between Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). The popular scientific notion of the time was that of Newton who believed that energy is simply the product of its mass and velocity (mv). Leibniz, however, argued that the energy factor was mv². Together with Voltaire (1694–1778), du Châtelet combined the theories of Leibniz and the experiments performed by Willem Gravesande (1688–1742) to show that the energy of a moving object is proportional to its mass and the square of its velocity (E ≈ mv²), thus rejecting the popular view of Newton. However, it did take about 150 more years before this idea gained true acceptance. Without the understanding that kinetic energy is proportional to the square of an object’s velocity, Einstein might not have correctly predicted that E = mc².

Gravesande’s experiment involved dropping heavy weights into soft clay. If Newton’s idea of kinetic energy (E = mv) was true, then a weight traveling twice as fast as a similar weight would sink into the clay twice as deep. However, what Gravesande discovered was that a small brass sphere traveling twice as fast as a similarly weighted sphere pushed in four times as far into the clay. A weight traveling three times as fast sank nine times as far into the clay.

The law of conservation of energy states that energy cannot be created or destroyed only converted between one form and another. A ball raised to a certain height above a tabletop has stored energy known as potential energy. The potential energy of an object raised above some reference height is calculated by multiplying the mass of the object times the acceleration due to gravity (9.81 m/s²) and the change in height of the object (PE = mgh). When the ball is released, its stored potential energy is converted into energy of motion, or kinetic energy. Kinetic energy is calculated by multiplying one-half the mass of the object by the square of its velocity (KE = ½mv²). The stored energy is converted completely into kinetic energy just as it passes the reference height (e.g., the ground, tabletop, top of the clay). Therefore, doubling the speed of an object increases its kinetic energy by a factor of four.

Materials

Safety Precautions

The clay used in this demonstration is considered nontoxic. Use caution and aim carefully when dropping the ball from great heights. Do not microwave the clay for more than 30-second intervals. Use only a laboratory-specific microwave (one not intended for food preparation). When using a hot plate to soften the clay, use only borosilicate glassware. Follow all laboratory safety guidelines. Wash hands thoroughly with soap and water after the demonstration.

Disposal

Save and store the clay in the zipper-lock bag for future use. To dispose of the clay, please consult your current Flinn Scientific Catalog/Reference Manual for general guidelines and specific procedures, and review all federal, state and local regulations that may apply, before proceeding. The clay may be disposed of according to Flinn Suggested Disposal Method #26a.

Prelab Preparation

1. Obtain two sticks of soft clay, two plastic semicircles and tape.
2. Align the two semicircles to form a complete ring and then use two small pieces of tape to secure them together (see Figure 1).
3. Place the ring onto a sheet of white paper or newspaper.
4. Knead two sticks of clay together to soften them up and make them uniform.
5. Place the clay into the ring to fill the ring. Add only enough clay to completely fill the ring until the clay is flush with the top edge of the ring (this will require approximately 1½ to 2 sticks of clay).
6. Press down firmly to remove any air pockets and to make sure the clay is uniformly packed inside the ring.
7. Press on the clay to make the top level with the edge of the ring.
8. Flip the ring over to make sure both sides of the clay are smooth and even with the edge of the ring.

1. Place the clay-filled ring into a large heavy-duty zipper-lock plastic bag, remove as much air as possible, and close the bag.
2. Prepare a hot water bath using a hot plate and a 1000-mL beaker half-full with water.
3. Heat the water to about 80 °C (keep the water below boiling).
4. Place the bag with the clay-filled ring into the hot water, making sure the bag is closed to prevent the clay from getting wet. Do not allow the plastic bag to touch the bottom of the beaker (the bag may melt from the heat of the hot plate).
5. Allow the clay to heat up for about 5 minutes.
6. Carefully remove the bag using tongs and place the bag on a paper towel.
7. Press on the clay to feel its firmness. It should be warm and pliable, and should easily form a crater when pressed. If the clay is soft and malleable, remove the clay-filled ring from the bag and place it on a white sheet of paper. If not, place the bag with the clay-filled ring back into the hot water bath for 2–3 more minutes and then test it again.
8. Once the clay has the appropriate firmness, place the clay-filled ring on a white sheet of paper.

1. Place the clay-filled ring into a 100-mm dia. borosilicate glass Petri dish.
2. Place the Petri dish and ring directly on a hot plate.
3. Turn on the hot plate to a low to medium setting.
4. Heat the Petri dish and clay-filled ring for five to seven minutes.
5. Using tongs or heat-resistant gloves, carefully flip the clay-filled ring up-side down to heat the cool side. Heat for an additional three to five minutes.
6. Carefully remove the ring from the Petri dish, turn off the hot plate, and place the clay-filled ring on a white sheet of paper.

1. Place the clay-filled disk onto a paper towel and place in a laboratory microwave oven with a 600- to 1000-mL beaker half-full with water (to capture the reflected microwaves).
2. On the “high” setting, microwave the clay for 15–30 seconds. Do not heat for longer than 30-second intervals.
3. After microwaving, feel the pliability of the clay. It should be warm and pliable and easily form a crater when pressed.
4. If the clay is not pliable enough, microwave for an additional 15–30 seconds and repeat step 3.
5. Once the clay has the appropriate firmness, place the clay-filled ring on a white sheet of paper.

Procedure

1. Place the ring with the warm, pliable clay on the demonstration table. Make sure the surface of the clay is flat.
2. Hold a meter stick vertically with the zero-mark on the edge of the ring.
3. Hold the ball bearing at the 20-cm mark above the left-hand side of the clay (see Figure 2).
4. Release the ball into the clay. It should make an indentation in the clay.
5. Carefully twist and remove the ball from the clay without significantly changing the indentation size or shape.
6. Hold the meter stick vertically with the zero-mark on the opposite edge of the ring.
7. Hold the ball bearing at the 80-cm mark above the right-hand side of the clay (see Figure 3). Make sure the ball will not land in the same indentation as the first drop.
8. Release the ball. It should make a deeper indentation in the clay.
9. Carefully twist and remove the ball from the clay without significantly changing the indentation size or shape.
10. Show the two different indentations to the students. One should be clearly deeper than the other. Students should be convinced that one indentation is about four times as deep as the other. 

(Optional) To verify the velocity-squared proportion quantitatively, carefully remove the semi-circles from around the clay. Place the clay on a cutting board or surface. Use a sharp knife to cut the clay in a line connecting the bottoms of the two indentations (so the depths can be seen simultaneously when the clay halves are separated). Cut the clay carefully so the indentations do not deform remarkably. Measure the depths of the indentations with a meter stick or ruler.

11. To repeat the demonstration, flatten out the indentations in the clay as much as possible and then turn the ring and clay over to use the “bottom” side. This can be repeated as often as necessary until the clay begins to cool and harden again. The clay has a high heat capacity and will tend to stay warm and soft for about 15 minutes after it has been properly heated.

Student Worksheet PDF

12627_Student1.pdf

Teacher Tips

• This kit contains enough materials to perform this demonstration indefinitely. Store the clay in the zipper-lock bag provided to prevent if from drying out and hardening. A small amount of water can be added to the clay to soften it up if it does dry out.
• The microwave heating method tends to be the fastest and most convenient method to soften the clay. However, not all laboratories have readily accessible microwaves, so two other heating methods are provided.
• Practice the ball drop method a few times on the hard clay to get an idea of where the ball will land. It takes a steady hand to drop the ball into the clay-filled disk.
• A paper towel can be added to the bag to soak up any water that may leak into the bag. However, it may require more time to heat the clay (around 10 minutes).
• The Kinetic Energy Ball Drop Worksheet may also be copied and used during the discussion or as a post-demonstration assignment, if desired.
• An infamous quote by Voltaire about the brilliant Emilie du Châtelet was that she “was a great man whose only fault was being a woman.” Fortunately, opportunities for women have changed considerably since the time of the Renaissance.
• Voltaire’s given name was François-Marie Arouet. Voltaire was his “pen name.”
• This demonstration shows that kinetic energy is proportional to the square of the velocity in two ways. The more dramatic way is by comparing the indentation depths. However, the second way (as students may notice) is demonstrated by the heights from which the ball is released. The ball must be dropped from four times the height to reach twice the velocity. This agrees with the conservation of energy principle—in order for the ball to have four times the kinetic energy just before it hits the clay, it must also start with four times the potential energy.
  
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Answers to Questions

1. From what height was the ball dropped for the first drop? For the second drop?

The first drop was from a height of 20 cm and the second drop was from a height of 80 cm.

2. Compare the indentations made by the ball in the first versus the second drop. How much deeper was the indentation caused by the second drop compared to the first drop?

The indentation caused by the second drop is about four times as deep as the indentation caused by the first drop. The second indentation is also larger in diameter and more clay appears to be pushed out along the edges.

3. Use the potential energy equation and the conservation of energy principle to calculate the kinetic energy of the ball just before it hit the clay after it was dropped from both heights. Assume the mass of the ball is 20 g.

Students can determine the kinetic energy just before it hits the clay by calculating the initial potential energy (due to the conservation of energy principle).

20 g × (1 kg/1000 g) × 9.81 m/s² × 0.20 m = 0.04 J (First drop)

20 g × (1 kg/1000 g) × 9.81 m/s² × 0.80 m = 0.16 J (Second drop)

4. Calculate the approximate velocity of the ball just before it hits the clay for both drops.

5. The purpose of this demonstration was to compare the kinetic energy of a ball when its velocity was doubled. Explain in words why the height that the ball was dropped from was increased by a factor of four in order to demonstrate this principle.

The speed of the ball increases as the square-root of the height. Therefore, in order to increase the speed by a factor of 2, the height of the drop must be raised to a factor of 4.

References

http://www.pbs.org/wgbh/nova/einstein/ance-sq.html (accessed July 2018).

http://physicsweb.org/articles/world/17/6/2/1 (accessed July 2018).

http://www.pbs.org/wgbh/nova/einstein/about.html (accessed July 2018).