Training Physics Math Energy, Work, and Power
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Energy, Work, and Power

24 min Physics Math

The concepts of energy, work, and power give us a scalar alternative to the vector methods of Newton’s laws. Work is the transfer of energy that occurs when a force acts through a displacement, kinetic energy captures the energy of motion, and potential energy stores the energy of position in a gravitational or elastic field. The work–energy theorem and conservation of energy let you bypass complicated force analyses entirely for many problems. Power — the rate at which work is done — connects these ideas to real-world performance ratings of motors, engines, and athletes.

Energy, Work, and Power

Energy conservation is one of the most powerful problem-solving tools in physics — it converts complex motion problems into simple algebra.

Work-Energy Theorem

$$W = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$

Conservation of Mechanical Energy

$$KE_1 + PE_1 = KE_2 + PE_2$$

$$\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2$$

Example 1

A 2 kg ball is dropped from 20 m. Find its speed at the ground.

  1. $mgh = \frac{1}{2}mv^2$ → $v = \sqrt{2gh} = \sqrt{2(10)(20)} = 20$ m/s.
Example 2

A roller coaster (1000 kg) starts from rest at 30 m height. Find speed at 10 m height.

  1. $mg(30) = \frac{1}{2}mv^2 + mg(10)$ → $mg(20) = \frac{1}{2}mv^2$
  2. $v = \sqrt{2g(20)} = \sqrt{400} = 20$ m/s.
Power

$$P = \frac{W}{t} = Fv$$

Power is the rate of doing work, measured in watts (W).

Example 3

A motor lifts 500 kg by 12 m in 10 s. Find the power.

  1. $W = mgh = 500(10)(12) = 60{,}000$ J.
  2. $P = 60{,}000/10 = 6{,}000$ W = 6 kW.

Practice Problems

1. A 5 kg ball is dropped from 45 m. Find its speed at the ground.
2. How much work is done by a 100 N force over 8 m?
3. A 60 kg skier starts from rest at 25 m elevation. Speed at the bottom?
4. What power lifts 200 kg by 15 m in 5 s?
5. A 1 kg pendulum swings from 0.5 m height. Speed at lowest point?
6. KE of a 4 kg object at 10 m/s?
Show Answer Key

1. $v = \sqrt{2(10)(45)} = 30$ m/s

2. $W = 100 \times 8 = 800$ J

3. $v = \sqrt{2(10)(25)} \approx 22.4$ m/s

4. $P = 200(10)(15)/5 = 6{,}000$ W

5. $v = \sqrt{2(10)(0.5)} \approx 3.16$ m/s

6. $KE = \frac{1}{2}(4)(100) = 200$ J

⚡ Energy, Work & Power Calculator
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