Eksperimentti: hyppykorkeuden määrittäminen impulssilla
Introduction
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = \int F dt = \Delta p = m\Delta v} . Actually Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta v} is our takeoff speed because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1=0} , and we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta v = v_{0} = \frac{J}{m} = \frac1m \int F dt} . Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s = s_0 + v_0 t + \tfrac12 at^2} and thus we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = v_0 t - \tfrac12 gt^2} because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_0=0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=-g = -9.81m/s^2} . However, for the velocity we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = v_0 - gt} and at the maximum height we have that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=0} , and thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_0 = gt} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=\frac{v_0}{g}} . Combining these two we have
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} h &= v_0 t - \tfrac12 gt^2 \\ &= v_0 \tfrac{v_0}g - \tfrac12 g\left( \frac{v_0}g \right)^2 \\ &= \frac{v_0^2}{g} - \tfrac12 \frac{v_0^2}{g} \\ &= \frac{v_0^2}{2g} \\ &= \frac{J^2}{2gm^2} = \frac{1}{2g} \left( \frac Jm \right)^2 \end{align} }
Example
The example gives
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} m &= 880 N /9.81 = 89.7 kg \\ J &= 700 Ns - 89.7 kg \times 9.81 \times 0.2895 s = 700 Ns - 254.75 Ns = 445.25 Ns \end{align} }
and thus we have
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} h &= \frac{J^2}{2gm^2} \\ &= \frac{(445.25 Ns)^2}{2 \times 9.81 m/s^2 \times (89.7 kg)^2 } \\ &= \frac{198 247. 5625}{8046.09} \\ &= 24.64 m\\ \end{align} }
References
https://www.thehoopsgeek.com/the-physics-of-the-vertical-jump/
https://www.brunel.ac.uk/~spstnpl/LearningResources/VerticalJumpLab.pdf
