College Physics by Openstax Chapter 2 Problem 47

(a) Calculate the height of a cliff if it takes 2.35 s for a rock to hit the ground when it is thrown straight up from the cliff with an initial velocity of 8.00 m/s.

(b) How long would it take to reach the ground if it is thrown straight down with the same speed?

Solution:

Part A

Refer to the figure below.

The known values are: t=2.35 s t=2.35\:\text{s}; y=0 my=0\:\text{m}; v0=+8.00 m/sv_0=+8.00\:\text{m/s}; and a=−9.8 m/s2a=-9.8\:\text{m/s}^2

Based on the given values, the formula that we shall use is

y=y0+v0t+12at2y=y_0+v_0t+\frac{1}{2}at^2

Substituting the values, we have

y=y0+v0t+12at20 =y0+(8.00 m/s)(2.35 s)+12(−9.80 m/s2)(2.35 s)2y0=8.26 m  (Answer)\begin{align*} y & =y_0+v_0t+\frac{1}{2}at^2 \\ 0\: & =y_0+\left(8.00\:\text{m/s}\right)\left(2.35\:\text{s}\right)+\frac{1}{2}\left(-9.80\:\text{m/s}^2\right)\left(2.35\:\text{s}\right)^2 \\ y_0 & =8.26\:\text{m} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

Therefore, the cliff is 8.26 meters high.

Part B

Refer to the figure below

The knowns now are: y=0 my=0\:\text{m}; y0=8.26 my_0=8.26\:\text{m}; v0=−8.00 m/sv_0=-8.00\:\text{m/s}; and a=−9.80 m/s2a=-9.80\:\text{m/s}^2

Based on the given values, we can use the formula

y=y0+v0t+12at2y=y_0+v_0t+\frac{1}{2}at^2

Substituting the values, we have

y=y0+v0t+12at20 m=8.26 m+(−8.00 m/s)t+12(−9.80 m/s2)t24.9t2+8t−8.26=0\begin{align*} y & =y_0+v_0t+\frac{1}{2}at^2 \\ 0\:\text{m} & =8.26\:\text{m}+\left(-8.00\:\text{m/s}\right)t+\frac{1}{2}\left(-9.80\:\text{m/s}^2\right)t^2 \\ 4.9 t^2+8t-8.26 & =0 \\ \end{align*}

Using the quadratic formula to solve for the value of tt, we have

t=−8±(8)2−4(4.9)(−8.26)2(4.9)t=0.717 s  (Answer)\begin{align*} t &=\frac{-8\pm \sqrt{\left(8\right)^2-4\left(4.9\right)\left(-8.26\right)}}{2\left(4.9\right)} \\ t &=0.717\:\text{s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}
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