Q. 82 - Calculus

Question

In a science fiction novel, gravity on the planet XV-37 acts very differently than it does in our universe. An object dropped from a $1000$-foot building on planet XV-37 will have a downward gravitational acceleration of $a\left(t\right)=-6t$ feet per second per second after falling for t seconds. Use antiderivatives to find equations for the position and velocity of such a falling object. What might be the consequences of living on this strange planet?

Step-by-Step Solution

Answer

The equation of the position of the object is $v\left(t\right)=-3t^2+v_0$.

The equation of the velocity of the object is $s\left(t\right)=-t^3+\frac{1}{2}{v}_0{t}^2+1000$.

A person won't be able to stand on the surface of the planet and none of the object will fall on the ground, making it impossible to survive on the planet.

Step 1. Given information.

Consider the given question,

$a\left(t\right)=-6t$

Step 2. Use the anti derivation of the acceleration function.

Using the anti derivative of the acceleration function to find the velocity function,

$$\begin{gathered} v\left(t\right)=-\frac{6}{2}{t}^2+v_0 \\ v\left(t\right)=-3t^2+v_0 \end{gathered}$$

Using the anti derivative of v to find the position vector,

$$\begin{gathered} s\left(t\right)=-\frac{3}{3}{t}^3+\frac{v_0}{2}{t}^2+s_0 \\ s\left(t\right)=-t^3+\frac{v_0}{2}{t}^2+s_0 \end{gathered}$$

As the object is falling from the height of $1000$ ft building, then the position function is given below,

$s\left(t\right)=-t^3+\frac{1}{2}{v}_0{t}^2+1000$

Step 3. Write the consequences of living on the strange planet.

The acceleration of the object decreases with time. Then, as t increases the value of a will become very less, hence a person can't stand on the surface of the planet and none of the object will fall on the ground.

Therefore, we can say that survival becomes impossible on the planet.