Understanding of energy and power in motion is essential. This comprehensive guide delves deep into problem-solving techniques involving these concepts, focusing particularly on practical applications, such as vehicles moving under various conditions. We aim to enhance your analytical skills in applying these principles to determine instantaneous acceleration and the forces involved in motion.

## Energy Transformations

**Kinetic Energy:**Energy of motion.**Potential Energy:**Stored energy.**Example:**A car going uphill converts kinetic energy to potential energy.

Image courtesy of inspiritvr

## Power in Motion

**Definition:**Power is how fast work is done or energy is transferred.**Application:**In moving objects, power shows how quickly energy changes, crucial for understanding motion.

## Mathematics in Motion

**Work-Energy Principle:****Work Done:**Calculated by $W = Fd \cos(\theta)$ , where $\theta$ is the angle between force and displacement direction.**Energy Changes:**Work done changes an object's kinetic or potential energy.

## Calculating Power

**Formulas:**- $P = \frac{W}{\text{time taken}}$
- $P = Fv$ , where $v$ is velocity.

## Energy in Uphill Motion

**Gravitational Potential Energy:**$PE = mgh$ , where $h$ is elevation gain.**Kinetic Energy:**Changes with the car’s speed.

## Power for Uphill Driving

**Calculation:**Considers gravity and resistive forces like friction.

## Example Problems

### Example 1: **Calculating Work Done**

A car with a mass of 1000 kg climbs a 50m hill at a constant speed. Calculate the work done against gravity.

**Solution:**

**Calculating Gravitational Potential Energy Gain (PE):**

- Formula: $PE = mgh$
- Given Data: Mass (m) = 1000 kg, Acceleration due to gravity (g) = 9.81 m/s², Height (h) = 50 m.
- Calculation: $PE = 1000 \times 9.81 \times 50 = 490,500 \text{ Joules}.$

**Conclusion:** The work done against gravity, which is equal to the gain in potential energy, is 490,500 Joules.

This graph shows that the work done remains constant at 490,500 Joules throughout the time it takes for the car to climb the hill.

### Example 2: Power Calculation

If the car takes 120 seconds to climb the hill, calculate the average power output.

**Solution:**

**Using Work Done from Problem 1:**- Work done = 490,500 Joules.

**Calculating Power:**- Formula: $P = \frac{W}{\text{time}}$, where $W$ is work done and $\text{time}$ is the time taken.
- Time taken = 120 seconds.
- Calculation: $P = \frac{490,500}{120} = 4,087.5 \text{ Watts}.$

**Conclusion:** The average power output of the car while climbing the hill is 4,087.5 Watts.

This graph illustrates how power changes over time. After the initial spike (due to division by very small time values), the power levels out, indicating the average power output of the car during the hill climb, which is about 4,087.5 Watts.

**Example 3: Energy Conversion Analysis**

Analyze the energy transformations involved when a car's brakes are applied to maintain a constant speed while descending a hill.

**Solution:**

**Kinetic Energy Remains Constant:**- The car's constant speed indicates its kinetic energy remains unchanged.

**Potential Energy Decreases:**- As the car descends, it loses gravitational potential energy.

**Energy Transformation Through Braking:**- The decrease in potential energy is converted into heat energy via the brakes.
- The braking system dissipates energy, preventing an increase in the car's kinetic energy.

**Conclusion:** During the descent with constant speed, the car's kinetic energy is constant, its potential energy decreases, and the lost potential energy is transformed into heat by the braking system. This demonstrates the conversion of mechanical energy into thermal energy.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.