# It took a 500.0-newton ballerina a force of 250 joules to lift herself upward through the air. How high did she jump?

Feb 6, 2016

0.5 m

#### Explanation:

The question is not well made because 'joule' is an unit of energy and work, not of force. I guess the 500 newton ballerina is her weight and the "force of 250 joules" is the kinetic energy of the jump.

It is important to make the hypothesis that the jump is vertical.

By using the 2nd Newton's law

F = m·a ( m = ballerina mass)

where F is the gravity force,

F = - m·g ( g = gravity acceleration). Then, -g = a = ${d}^{2} \frac{x}{\mathrm{dt}} ^ 2$

doing the integration at both sides of equality

-g·t = ${v}_{f} - {v}_{0}$

When the ballerina is in the highest point of her jump ${v}_{f} = 0$ => g· t = ${v}_{0}$ => the time of jump is t = ${v}_{0}$ / g

doing another integration at both sides of equality

${t}^{2} / 2$ = h and replacing here the value of t

h = ${v}_{0}^{2} / \left(2 g\right)$
The value of kinetic energy is ${E}_{k} = m \cdot {v}_{0}^{2} / 2$ then ${v}_{0}^{2} = 2 \cdot {E}_{k} / m$ and replacing it in the previous formula

$h = {E}_{k} / \left(m \cdot g\right)$ but $m \cdot g$ is the ballerina's weight

$h = {E}_{k} / W =$ 250J/500N = 0.5 m .