An unwary football player collides with a padded goalpost while running at a velocity of 7.50 m/s and comes to a full stop after compressing the padding and his body 0.350 m.
a) What is his deceleration?
b) How long does the collision last?
Solution:
We are given the following: v_0=7.50\:\text{m/s}; v_f=0.00\:\text{m/s}; and\Delta x=0.350\:\text{m}.
Part A
We are going to use the formula
\left(v_f\right)^2=\left(v_0\right)^2+2a\Delta x
Solving for the acceleration a in terms of the other variables:
a=\frac{\left(v_f\right)^2-\left(v_0\right)^2}{2\Delta x}Substituting the given values:
\begin{align*}
a & =\frac{\left(v_f\right)^2-\left(v_0\right)^2}{2\Delta x} \\
a & =\frac{\left(0\:\text{m/s}\right)^2-\left(7.50\:\text{m/s}\right)^2}{2\left(0.350\:\text{m}\right)} \\
a & =-80.4\:\text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}Part B
We are going to use the formula
\Delta x=v_{ave}tSince v_{ave}=\frac{v_f+v_0}{2}, we can write the formula as
\Delta x=\frac{v_f+v_0}{2}\cdot tSolving for time t in terms of the other variables:
\:t=\frac{2\Delta x}{v_f+v_0}Substituting the given values:
\begin{align*}
t&=\frac{2\Delta x}{v_f+v_0} \\
t&=\frac{2\left(0.350\:\text{m}\right)}{0\:\text{m/s}+7.50\:\text{m/s}} \\
t& =9.33\times 10^{-2}\:\text{s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}
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