To analyze the kinetic energy of the balls from the data provided in the table, we will utilize the kinetic energy formula:
\[ KE = \frac{1}{2} m v^2 \]
Where:
- \( KE \) is the kinetic energy,
- \( m \) is the mass of the object, and
- \( v \) is the velocity of the object.
We can confirm the kinetic energy values provided in the table using this formula:
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For Ball 1: \[ KE = \frac{1}{2} \cdot 5 \cdot 3^2 = \frac{1}{2} \cdot 5 \cdot 9 = 22.5 , J \] (matches)
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For Ball 2: \[ KE = \frac{1}{2} \cdot 4 \cdot 3^2 = \frac{1}{2} \cdot 4 \cdot 9 = 18.0 , J \] (matches)
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For Ball 3: \[ KE = \frac{1}{2} \cdot 3 \cdot 3^2 = \frac{1}{2} \cdot 3 \cdot 9 = 13.5 , J \] (matches)
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For Ball 4: \[ KE = \frac{1}{2} \cdot 5 \cdot 6^2 = \frac{1}{2} \cdot 5 \cdot 36 = 90.0 , J \] (matches)
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For Ball 5: \[ KE = \frac{1}{2} \cdot 5 \cdot 4^2 = \frac{1}{2} \cdot 5 \cdot 16 = 40.0 , J \] (matches)
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For Ball 6: \[ KE = \frac{1}{2} \cdot 5 \cdot 5^2 = \frac{1}{2} \cdot 5 \cdot 25 = 62.5 , J \] (matches)
Given the calculations confirm the kinetic energy values, we can analyze the relationships and trends that may be expected in graphical representations of the data.
- Mass vs. Kinetic Energy: Since kinetic energy increases as mass increases (for constant velocities) and is also affected by the square of the velocity, we expect to see a relationship here.
- Velocity vs. Kinetic Energy: As velocity increases, the kinetic energy increases with the square of the velocity. Thus we expect to see a nonlinear relationship where higher speeds result in significantly greater kinetic energies.
Based on these attributes, you might expect to see:
- A linear or slightly increasing graph for mass vs. kinetic energy if velocities are constant.
- A quadratic graph (parabolic) for velocity squared vs. kinetic energy, indicating that as the square of the velocity increases, so does the kinetic energy substantially.
If you have specific pairs of graphs in mind, look for one graph depicting the mass of the balls against their kinetic energy, and another graph depicting the square of the velocity against kinetic energy for the best representation of the data provided.