Eksperimentti: hyppykorkeuden määrittäminen impulssilla

Introduction

Force exerted on the force plate

Jumping on the force plate you can feel the force. We use time of flight method to estimate the height of the jump.

Theory

Impulse $J=\int Fdt=\Delta p=m\Delta v$. Actually $\Delta v$ is our takeoff speed because $v_{1}=0$, and we have $\Delta v=v_{0}={\frac {J}{m}}={\frac {1}{m}}\int Fdt$. Because $s=s_{0}+v_{0}t+{\tfrac {1}{2}}at^{2}$ and thus we have $h=v_{0}t-{\tfrac {1}{2}}gt^{2}$ because $h_{0}=0$ and $a=-g=-9.81m/s^{2}$. However, for the velocity we have $v=v_{0}-gt$ and at the maximum height we have that $v=0$, and thus $v_{0}=gt$ and $t={\frac {v_{0}}{g}}$. Combining these two we have

${\begin{aligned}h&=v_{0}t-{\tfrac {1}{2}}gt^{2}\\&=v_{0}{\tfrac {v_{0}}{g}}-{\tfrac {1}{2}}g\left({\frac {v_{0}}{g}}\right)^{2}\\&={\frac {v_{0}^{2}}{g}}-{\tfrac {1}{2}}{\frac {v_{0}^{2}}{g}}\\&={\frac {v_{0}^{2}}{2g}}\\&={\frac {J^{2}}{2gm^{2}}}\end{aligned}}$

Example

The example gives Failed to parse (syntax error): {\displaystyle \begin{align*} m &= 880 N /9.81 = 89.7 kg \\ J &= 900 Ns \end{align*} }

and thus we have

Failed to parse (syntax error): {\displaystyle \begin{align*} h &= \frac{J^2}{2gm^2} \\ &= \frac{(900 Ns)^2}{2 \times 9.81 m/s^2 \times (89.7 kg)^2 } \\ &= \frac{810000}{8046.09} \\ &= 100.67 m\\ \end{align*} }

References

https://www.thehoopsgeek.com/the-physics-of-the-vertical-jump/

https://www.brunel.ac.uk/~spstnpl/LearningResources/VerticalJumpLab.pdf