Chapter 1.5, Problem 39PS

Solution

To find: time it takes for the rock to hit the surface for different planets.

For Earth: $t\approx 3.06\text{ sec}$ .

For Mars: $t=5\text{ sec}$ .

For Jupiter: $t\approx 1.99\text{ sec}$ .

For Saturn: $t\approx 3.16\text{ sec}$ .

For Pluto: $t\approx 12.25\text{ sec}$ .

Given:

On any planet, the height h (in feet) of a falling object t seconds after it is dropped can be modeled after by the function:

  $h\left(t\right)=-\frac{g}{2}{t}^2+h_0$ .

Where $h_0$ is the object’s initial height (in feet) and g is the acceleration (in feet per second squared).

The rock is dropped from a height of 150 feet.

The acceleration for different planets is given in the table below:

Planet	Earth	Mars	Jupiter	Saturn	Pluto
$g\left(ft/{\sec }^2\right)$	32	12	76	30	2

Property Used:

  $\begin{array}{c}\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}\\ \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\end{array}$

Calculation:

By the height function, the height of the object h (in feet) above the ground t seconds after it is dropped is given by:

  $h\left(t\right)=-\frac{g}{2}{t}^2+h_0$

Now, rock is dropped from a height of 150 feet so, its initial height is 150 feet.

So, $h_0=150$ .

Thus, the equation giving the height of the object h (in feet) above the ground t seconds after it is dropped is:

  $h\left(t\right)=-\frac{g}{2}{t}^2+150$ .

Now, when the rock hits the surface, its height becomes 0. So, target is to determine t for which $h\left(t\right)=0$ .

  $0=-\frac{g}{2}{t}^2+150$

This is equivalent to,

  $\frac{g}{2}{t}^2=150$ .

Divide both sides by $\frac{g}{2}$ ,

  $t^2=\frac{300}{g}$ .

Take square root of both sides,

  $t=\pm \sqrt{\frac{300}{g}}$ .

Since, t represents time it is a non negative quantity. Thus, the solution is $t=\sqrt{\frac{300}{g}}$ .

Now, first let the planet is Earth and for Earth $g=32ft/{\sec }^2$ , so time taken by rock to hit the ground is:

  $\begin{array}{c}t=\sqrt{\frac{300}{g}}\\ =\sqrt{\frac{300}{32}}\\ \approx 3.06\text{ sec}\end{array}$

Now, first let the planet is Mars and for Mars $g=12ft/{\sec }^2$, so time taken by rock to hit the ground is:

  $\begin{array}{c}t=\sqrt{\frac{300}{g}}\\ =\sqrt{\frac{300}{12}}\\ =5\text{ sec}\end{array}$

Now, first let the planet is Jupiter and for Jupiter $g=76ft/{\sec }^2$, so time taken by rock to hit the ground is:

  $\begin{array}{c}t=\sqrt{\frac{300}{g}}\\ =\sqrt{\frac{300}{76}}\\ \approx 1.99\text{ sec}\end{array}$

Now, first let the planet is Saturn and for Saturn $g=30ft/{\sec }^2$, so time taken by rock to hit the ground is:

  $\begin{array}{c}t=\sqrt{\frac{300}{g}}\\ =\sqrt{\frac{300}{30}}\\ \approx 3.16\text{ sec}\end{array}$

Now, first let the planet is Pluto and for Pluto $g=2ft/{\sec }^2$, so time taken by rock to hit the ground is:

  $\begin{array}{c}t=\sqrt{\frac{300}{g}}\\ =\sqrt{\frac{300}{2}}\\ \approx 12.25\text{ sec}\end{array}$