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When an object undergoes a change in kinetic energy over a period of time, the average power required to achieve this change can be determined. Average power can be seen as the overall rate of energy transfer or consumption during a given time frame. In mathematical terms, average power (\( P_{\text{avg}} \)) is calculated by dividing the work done (\( W \)) by the time (\( t \)) taken, expressed as:

• Average power formula: \( P_{\text{avg}} = \frac{W}{t} \).

In this exercise, the car's kinetic energy at the end of 30 seconds is 300,000 Joules. Since the automobile takes 30 seconds to accelerate from rest, the average power can be calculated by dividing the change in energy by 30 seconds:

• \( P_{\text{avg}} = \frac{300,000 \text{ J}}{30 \text{ s}} = 10,000 \text{ W}\), or 10 kW.

This value represents the steady rate at which energy has been used by the car to achieve its final velocity.

Instantaneous power refers to the amount of power being used at a particular moment in time. It provides a snapshot of energy consumption at any given instant, unlike average power which is spread across the entire duration. The equation for instantaneous power (\( P \)) in situations concerning motion is given by the product of force (\( F \)) and velocity (\( v \)):

• Instantaneous power formula: \( P = Fv \)

To find the instantaneous power of the car at the end of the 30-second interval, first determine the acceleration (\( a \)) and apply:

• Acceleration: \( a = \frac{v}{t} = \frac{20 \text{ m/s}}{30 \text{ s}} = \frac{2}{3} \text{ m/s}^2 \)
• Force: \( F = m \cdot a = 1500 \text{ kg} \times \frac{2}{3} \text{ m/s}^2 = 1000 \text{ N} \)
• Instantaneous power: \( P = 1000 \text{ N} \times 20 \text{ m/s} = 20,000 \text{ W} \)

Thus, the instantaneous power at the end of 30 seconds is 20,000 Watts or 20 kW, indicating the much higher energy usage at that specific moment compared to the average.

Acceleration is the rate at which an object's velocity changes over time. It's a fundamental concept in physics that describes how quickly an object speeds up or slows down. Constant acceleration implies that the object is gaining velocity at a uniform rate. For any object starting from rest (initial velocity \( u = 0 \)), the simplest form to find acceleration (\( a \)) when final velocity (\( v \)) and time (\( t \)) are known, is through the formula:

• Acceleration formula: \( a = \frac{v - u}{t} \)

In this exercise, the car accelerates uniformly from rest to a velocity of 20 m/s over a time interval of 30 seconds:

• Substituting values, \( a = \frac{20 \text{ m/s}}{30 \text{ s}} = \frac{2}{3} \text{ m/s}^2 \)

This tells us that every second, the car's speed increases by \( \frac{2}{3} \text{ m/s} \), translating the car's consistent gain in velocity during the 30 seconds. Understanding acceleration is crucial to calculate forces acted on any object and thus understand power requirements and energy utilization in physical systems.