# Energy and Conservation Laws

Mathematics I · Physics · https://tryals.app/learn/mathematics-i/energy-and-conservation-laws

## The Currency of Physics

**Energy** is a conserved quantity that flows through every physical process, the universal currency of physics. **Work** is the transfer of energy by a force acting over a distance, measured in **joules**.

| Case | Formula | Key Rule |
| --- | --- | --- |
| Constant Force | $W = Fd\cos\theta$ | Perpendicular force does zero work |
| Kinetic Energy | $KE = \tfrac{1}{2}mv^2$ | Doubling speed **quadruples** $KE$ |
| Gravitational PE | $PE = mgh$ | Stored energy near Earth |
| Spring PE | $PE = \tfrac{1}{2}kx^2$ | Stored energy in a spring |

The **work–energy theorem** states that net work equals the change in kinetic energy. **Power** is the **rate** of doing work, $P = \frac{dW}{dt}$, measured in **watts**.

## Conservation Laws

The crown jewel is **conservation of energy**: in an isolated system with only conservative forces, total mechanical energy $KE + PE$ is **constant**. Energy converts between forms but is never created or destroyed.

- **Free fall**: trades potential energy for kinetic energy.
- **Friction**: converts mechanical energy into heat.
- **System bookkeeping**: lets you relate initial and final states without tracking detailed motion.

> **Common pitfall:** Confusing **energy** with **power**, and forgetting the **square** in $KE = \tfrac{1}{2}mv^2$. Doubling speed quadruples $KE$, which is why stopping distances grow so sharply.

## Practice questions

3 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. Doubling a car's speed roughly **quadruples** its braking distance. Which relationship explains that, and why does it matter for speed limits?

A. Braking force halves at higher speeds, doubling the distance twice over
B. Air resistance rises with v², which lengthens the stop
C. Reaction time doubles with speed, contributing the extra factor
D. Kinetic energy goes as v², so twice the speed is four times the energy

**Answer:** D. Kinetic energy goes as v², so twice the speed is four times the energy

**Why:** Energy scales with the square of speed, and the brakes must dissipate all of it as work over the stopping distance, so 30 to 60 km/h is four times the distance, not twice. It is the strongest argument for urban speed limits, and it is entirely contained in the exponent.

Page: https://tryals.app/practice/mathematics-i/energy-and-conservation-laws/doubling-a-cars-speed-roughly-quadruples-its-braking-distance-which

### 2. Gravity does no work on a satellite in a circular orbit.

**Answer:** True

**Why:** True. The gravitational force points to the centre while the velocity is tangential, perpendicular at every instant, so $W = Fd\cos 90° = 0$ and the speed never changes. In an *elliptical* orbit they are not perpendicular, which is exactly why such a satellite speeds up near perigee.

Page: https://tryals.app/practice/mathematics-i/energy-and-conservation-laws/gravity-does-no-work-on-a-satellite-in-a-circular-orbit

### 3. Order the energy argument that finds the landing speed of a ball dropped from a height h.

**Answer:**

1. At the moment of release all the energy is potential: PE = mgh and KE = 0
2. As it falls, potential energy converts to kinetic energy with the sum unchanged
3. At the ground all the energy is kinetic, so set mgh equal to one half m v squared
4. Cancel the mass from both sides
5. Solve to get v equal to the square root of 2gh, independent of the mass

**Why:** Conservation of energy connects the initial and final states directly, with no need for the equations of motion. The cancellation of m is the famous result that all objects fall alike in the absence of air resistance. Galileo's conclusion, recovered in four lines of bookkeeping.

Page: https://tryals.app/practice/mathematics-i/energy-and-conservation-laws/order-the-energy-argument-that-finds-the-landing-speed-of-a-ball