The table shows the kinetic energy of six balls rolling on the ground.

Question

(Ball) 1, 2, 3, 4, 5, 6 (Velocity) (m/s) 3, 3, 3, 6, 4, 5 (Mass) (kg) 5, 4, 3, 5, 5, 5 (Kinetic Energy) (J) 22.5, 18.0, 13.5, 90.0, 40.0, 62.5

Which pair of graphs is best supported by the data in the table?
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Answer 1

To analyze the kinetic energy of the balls, we can use the formula for kinetic energy (KE), which is given by:

\[ \text{KE} = \frac{1}{2} mv^2 \]

Where:

• \( m \) is the mass of the object (in kg)
• \( v \) is the velocity of the object (in m/s)

Given the data for the balls, we can calculate the expected kinetic energy for each ball and compare it to the provided values:

1. Ball 1:

   • Velocity = 3 m/s
   • Mass = 5 kg
   • KE = \( \frac{1}{2} \times 5 \times (3)^2 = \frac{1}{2} \times 5 \times 9 = 22.5 , J \) [Matches the data]

2. Ball 2:

   • Velocity = 3 m/s
   • Mass = 4 kg
   • KE = \( \frac{1}{2} \times 4 \times (3)^2 = \frac{1}{2} \times 4 \times 9 = 18.0 , J \) [Matches the data]

3. Ball 3:

   • Velocity = 3 m/s
   • Mass = 3 kg
   • KE = \( \frac{1}{2} \times 3 \times (3)^2 = \frac{1}{2} \times 3 \times 9 = 13.5 , J \) [Matches the data]

4. Ball 4:

   • Velocity = 6 m/s
   • Mass = 5 kg
   • KE = \( \frac{1}{2} \times 5 \times (6)^2 = \frac{1}{2} \times 5 \times 36 = 90.0 , J \) [Matches the data]

5. Ball 5:

   • Velocity = 4 m/s
   • Mass = 5 kg
   • KE = \( \frac{1}{2} \times 5 \times (4)^2 = \frac{1}{2} \times 5 \times 16 = 40.0 , J \) [Matches the data]

6. Ball 6:

   • Velocity = 5 m/s
   • Mass = 5 kg
   • KE = \( \frac{1}{2} \times 5 \times (5)^2 = \frac{1}{2} \times 5 \times 25 = 62.5 , J \) [Matches the data]

The calculations confirm that the kinetic energy values listed in the table are accurate based on the corresponding masses and velocities.

In terms of graphing this data, two potential graphs could illustrate the relationship between mass, velocity, and kinetic energy:

1. Graph of Kinetic Energy vs. Mass (for constant velocities): This would show a quadratic relationship because kinetic energy is proportional to the mass and the square of the velocity.

2. Graph of Kinetic Energy vs. Velocity (for constant masses): This would also show a quadratic relationship since KE is proportional to the square of the velocity.

Thus, a pair of graphs that could best illustrate this data would be:

Answer: Kinetic Energy vs. Mass (for constant velocities) and Kinetic Energy vs. Velocity (for constant masses).