Question

Why is the following situation impossible? A normally proportioned adult walks briskly along a straight line in the$\mathbf{+}\mathit{x}$direction, standing straight up and holding his right arm vertical and next to his body so that the arm does not swing. His right hand holds a ball at his side a distance$\mathit{h}$above the floor. When the ball passes above a point marked as$\mathit{x}\mathbf{=}\mathbf{0}$on the horizontal floor, he opens his fingers to release the ball from rest relative to his hand. The ball strikes the ground for the first time at position$\mathit{x}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{00}\mathit{h}$

Short answer

The situation is impossible because a person cannot walk with a speed of $68.3\mathrm{km}/\mathrm{h}$.

Step-by-step solution

The time formula.

Write the formula to calculate time taken by ball to hit the floor

$\mathit{t}\mathbf{=}\sqrt{\frac{\mathbf{2}\mathbf{h}}{\mathbf{g}}}$

Here,$\mathbf{h}$is the vertical distance above the floor and$\mathit{g}$is the acceleration due to gravity.

The comparison between man and ball speed.

A normally proportioned adult walks along a straight line in the$+x$direction and his right hand hold a ball at his side a distance$h$above the floor.

Write the formula to calculate time taken by ball to hit the floor

$t=\sqrt{\frac{2h}{g}}$

Here, $h$is the vertical distance above the floor and $g$is the acceleration due to gravity.

Write the formula to calculate distance covered by ball

$x=vt$

Here,$x$is the distance covered by ball is and$v$is velocity of the ball.

Substitute$\sqrt{\frac{2h}{g}}$for$t$in above equation.

$x=v\sqrt{\frac{2h}{g}}$

In the given condition is $x=7.00h$. Substitute $7.00h$for $x$in the above equation.

$\begin{array}{c}7.00h=v\sqrt{\frac{2h}{g}}\\ v=7.00\left(\sqrt{\frac{gh}{2}}\right)\end{array}$

Normally a person's height is$1.5\text{m}$so substitute$1.5\text{m}$for$h$and$9.81\mathrm{m}/{\mathrm{s}}^2$for$g$to find.

role="math" localid="1663602943238" $\begin{array}{c}v=7.00h\left(\sqrt{\frac{9.81\mathrm{m}/{\mathrm{s}}^2}{2h}}\right)\\ =7.00\left(\sqrt{\frac{1.5\mathrm{m}\times 9.81\mathrm{m}/{\mathrm{s}}^2}{2}}\right)\\ =(18.98\mathrm{m}/\mathrm{s})\left(\frac{3.6\mathrm{km}/\mathrm{h}}{1\mathrm{m}/\mathrm{s}}\right)\\ =68.3\mathrm{km}/\mathrm{h}\end{array}$

This speed is very high and cannot achieved by a person. So, it is impossible for the ball to strike for the first time at position role="math" localid="1663602967708" $x=7.00h$.