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

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

Note that $J/m=v_{0}$, and thus the equation gives the correct equation.

Example

The example gives

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}m&=880N/9.81=89.7kg\\J&=464Ns-89.7kg\times 9.81\times 0.2895s=464Ns-254.75Ns=209.25Ns\end{aligned}}}

and thus we have

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}h&={\frac {J^{2}}{2gm^{2}}}\\&={\frac {(209.25Ns)^{2}}{2\times 9.81m/s^{2}\times (89.7kg)^{2}}}\\&={\frac {43785.5625}{315728.5716}}\\&=0.139m\\\end{aligned}}}

The takeoff velocity is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v_{0}={\frac {J}{m}}={\frac {209.25Ns}{89.7kg}}=2.33m/s} .

References

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

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