College Physics by Openstax Chapter 2 Problem 33

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: v0=7.50 m/sv_0=7.50\:\text{m/s}; vf=0.00 m/sv_f=0.00\:\text{m/s}; andΔx=0.350 m\Delta x=0.350\:\text{m}.

Part A

We are going to use the formula

(vf)2=(v0)2+2aΔx\left(v_f\right)^2=\left(v_0\right)^2+2a\Delta x

Solving for the acceleration aa in terms of the other variables:

a=(vf)2−(v0)22Δxa=\frac{\left(v_f\right)^2-\left(v_0\right)^2}{2\Delta x}

Substituting the given values:

a=(vf)2−(v0)22Δxa=(0 m/s)2−(7.50 m/s)22(0.350 m)a=−80.4 m/s2  (Answer)\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

Δx=vavet\Delta x=v_{ave}t

Since vave=vf+v02v_{ave}=\frac{v_f+v_0}{2}, we can write the formula as

Δx=vf+v02⋅t\Delta x=\frac{v_f+v_0}{2}\cdot t

Solving for time tt in terms of the other variables:

 t=2Δxvf+v0\:t=\frac{2\Delta x}{v_f+v_0}

Substituting the given values:

t=2Δxvf+v0t=2(0.350 m)0 m/s+7.50 m/st=9.33×10−2 s  (Answer)\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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