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A student drops a ball three different times.

The table shows the ball's kinetic energy upon hitting the ground from different heights.
Height of Drop (meters) | Kinetic Energy (joules)
1.0 | 48
1.5 | 72
2.0 | 96
The student claims that a ball dropped 3.0 meters would have fewer than 86 joules of kinetic energy upon hitting the ground. Is she correct? Why?
A. She is wrong. The higher the drop, the less kinetic the ball will have upon hitting the ground.
B. She is right! The higher the drop, the less kinetic energy the ball will have upon hitting the ground.
C. She is wrong. The higher the drop, the more kinetic energy the ball will have upon hitting the ground.
D. She is right! The higher the drop, the greater kinetic the ball will have upon hitting the ground.

3 answers

To determine whether the student's claim is correct, we need to analyze the relationship between the height of the drop and the ball's kinetic energy upon hitting the ground.

From the given data, we can observe the following:

  • At a height of 1.0 meters, the kinetic energy is 48 joules.
  • At a height of 1.5 meters, the kinetic energy is 72 joules.
  • At a height of 2.0 meters, the kinetic energy is 96 joules.

We notice that as the height of the drop increases, the kinetic energy increases. This suggests a direct relationship between height and kinetic energy.

We can calculate the kinetic energy for a height of 3.0 meters by looking for a pattern. From the data:

  • The increase from 1.0 m to 1.5 m (0.5 m increase) results in an increase of 24 joules in kinetic energy (72 - 48).
  • The increase from 1.5 m to 2.0 m (0.5 m increase) results in an increase of 24 joules in kinetic energy (96 - 72).

This indicates that for every 0.5 meter increase in height, the kinetic energy increases by 24 joules.

Following this pattern:

  • For 2.0 m to 2.5 m (0.5 m increase), we can expect another increase of 24 joules: 96 + 24 = 120 joules.
  • Then, for 2.5 m to 3.0 m (another 0.5 m increase), we expect another increase of 24 joules: 120 + 24 = 144 joules.

Thus, we can conclude that at a height of 3.0 meters, the kinetic energy would be 144 joules, which is well above 86 joules.

Given this analysis, the student's claim that a 3.0-meter drop would have fewer than 86 joules of kinetic energy is incorrect.

Therefore, the correct answer is: C. She is wrong. The higher the drop, the more kinetic energy the ball will have upon hitting the ground.

are you sure this is correct

Yes, let's break it down step by step again to confirm the analysis:

  1. Understanding the Relationship: The kinetic energy of an object as it hits the ground is derived from the potential energy it has at its height. The formula for gravitational potential energy (which converts to kinetic energy at the moment just before hitting the ground) is given by: \[ PE = mgh \] Where:

    • \(PE\) = potential energy (which equals kinetic energy upon hitting ground)
    • \(m\) = mass of the ball (constant for this analysis)
    • \(g\) = acceleration due to gravity (approximately 9.81 m/s²)
    • \(h\) = height from which the ball is dropped

  2. Given Data:

    • Height of 1.0 meters corresponds to 48 joules of kinetic energy.
    • Height of 1.5 meters corresponds to 72 joules.
    • Height of 2.0 meters corresponds to 96 joules.

  3. Calculating the Rate of Increase in Kinetic Energy:

    • The difference in kinetic energy between 1.0 m and 1.5 m: \[ 72 \text{ J} - 48 \text{ J} = 24 \text{ J} \]
    • The difference between 1.5 m and 2.0 m: \[ 96 \text{ J} - 72 \text{ J} = 24 \text{ J} \]

From these observations, we see a pattern that for every increase of 0.5 meters in height, there is an increase of 24 joules in kinetic energy.

  1. Extrapolating to Higher Heights:

    • Using the pattern:
      • Adding 24 joules for each additional 0.5 meters from 2.0 meters:
        • Height of 2.5 meters: \[ 96 \text{ J} + 24 \text{ J} = 120 \text{ J} \]
        • Height of 3.0 meters: \[ 120 \text{ J} + 24 \text{ J} = 144 \text{ J} \]

  2. Conclusion: Given that the pattern indicates that the kinetic energy at a height of 3.0 meters is 144 joules, which is significantly greater than 86 joules, the student's claim that a 3.0-meter drop would result in fewer than 86 joules is indeed incorrect.

So, reaffirming, the correct answer remains: C. She is wrong. The higher the drop, the more kinetic energy the ball will have upon hitting the ground.

If you have any further questions or need more clarification, feel free to ask!