Work and Energy
Beginner
mechanical energy and conservation
Source: High school physics (Chinese)
Problem Sets:
work/energy basic
work-energy 1018
Problem
An object with a mass of $m = 1$ kg is in free fall. Take the acceleration due to gravity as $g = 10$ m/s².
- What is the decrease in its gravitational potential energy during the first second of fall?
- What is the decrease in its gravitational potential energy during the second second of fall?
\Delta E_{p1} = 50 J \Delta E_{p2} = 150 J
The decrease in gravitational potential energy is equal to the work done by gravity, $\Delta E_p = W_g = mg\Delta h$. The distance fallen from rest is given by $\Delta h = \frac{1}{2}gt^2$.
For the first second (from $t=0$ s to $t=1$ s):
$$\Delta h_1 = \frac{1}{2}g(1)^2 - \frac{1}{2}g(0)^2 = \frac{1}{2}g$$ $$\Delta E_{p1} = mg\Delta h_1 = \frac{1}{2}mg^2$$ $$\Delta E_{p1} = \frac{1}{2}(1 \text{ kg})(10 \text{ m/s}^2)^2 = 50 \text{ J}$$For the second second (from $t=1$ s to $t=2$ s):
$$\Delta h_2 = \frac{1}{2}g(2)^2 - \frac{1}{2}g(1)^2 = \frac{3}{2}g$$ $$\Delta E_{p2} = mg\Delta h_2 = \frac{3}{2}mg^2$$ $$\Delta E_{p2} = \frac{3}{2}(1 \text{ kg})(10 \text{ m/s}^2)^2 = 150 \text{ J}$$