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Energy, Work, and Power

24 min Physics Math

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.

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

Example 2

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

$mg(30) = \frac{1}{2}mv^2 + mg(10)$ → $mg(20) = \frac{1}{2}mv^2$

$v = \sqrt{2g(20)} = \sqrt{400} = 20$ 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.

$W = mgh = 500(10)(12) = 60{,}000$ J.

$P = 60{,}000/10 = 6{,}000$ 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