Question

A stone is thrown vertically into the air at an initial velocity of 96 ft/s. On Mars, the height s (in feet) of the stone above the ground after t seconds is s=69t-6t2and on Earth it is s=69t-16t2. How much higher will the stone travel on Mars than on Earth?

Answer 1

240 ft

Step-by-step explanation:

t = Time taken

u = Initial velocity = 96 ft/s

v = Final velocity

s = Displacement

a = Acceleration = 12 m/s² on Mars 32 ft/s² on Earth negative due to upward direction

Mars

`s=ut+(1)/(2)at^2\\\Rightarrow s=96* t+(1)/(2)* -12* t^2\\\Rightarrow s=96t-6t^2\ ft`

Earth

`s=ut+(1)/(2)at^2\\\Rightarrow s=96* t+(1)/(2)* -32* t^2\\\Rightarrow s=96t-16t^2\ ft`

Differentiating the first equation with respect to time we get

`(ds)/(dt)=96-12t`

Equating with zero

`0=96-12t\\\Rightarrow t=(96)/(12)=8\ s`

Differentiating the second equation with respect to time we get

`(ds)/(dt)=96-32t`

Equating with zero

`0=96-32t\\\Rightarrow t=(96)/(32)=3\ s`

Applying the time taken to the above equations, we get

`s=96t-6t^2\ ft\\\Rightarrow s=96* 8-6* 8^2\\\Rightarrow s=384`

`s=96t-16t^2\\\Rightarrow s=96* 3-16* 3^2\\\Rightarrow s=144`

Difference in height = 384-144 = 240 ft

The stone will travel 240 ft higher on Mars