Final answer:
The average rate of change in kinetic energy as the velocity increases from 5 m/s to 12.5 m/s is calculated using the difference in kinetic energies at these velocities, resulting in an average rate of 175 joules per m²/s.
Step-by-step explanation:
The calculation of the average rate of change in kinetic energy for an object as its velocity increases from 5 m/s to 12.5 m/s involves using the given difference quotient of K(v), which is 20v + 10h. To find the average rate of change, we plug in the initial and final velocities (v1 = 5 m/s and v2 = 12.5 m/s) into the difference quotient and solve for the change in kinetic energy over this interval.
Using the formula K(v) = (1/2)mv², where m is mass, and v is velocity, we can write:
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- K(5) = 0.5 × 20 kg × (5 m/s)² = 250 J
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- K(12.5) = 0.5 × 20 kg × (12.5 m/s)² = 1562.5 J
The change in kinetic energy (ΔKE) is K(12.5) - K(5) = 1562.5 J - 250 J = 1312.5 J. The change in velocity (Δv) is 12.5 m/s - 5 m/s = 7.5 m/s.
Thus, the average rate of change is:
average rate = ΔKE / Δv = 1312.5 J / 7.5 m/s = 175 J/m/s
The correct answer for the average rate of change in kinetic energy as the velocity increases from 5 m/s to 12.5 m/s is 175 joules per meter per second (J/m/s), which converts to 175 joules per m²/s considering the units of velocity squared in the kinetic energy formula.