Q. 07 - Calculus

Question

A falling object accelerates downwards due to gravity at a rate of 9.8 meters per second squared. Suppose an object is dropped and falls to the ground from a height of 100 meters.

 (a) Set up and solve a first-order initial-value problem whose solution is the velocity v(t) of the object at time t. 

(b) Use your answer to part (a) to set up and solve a first-order initial-value problem whose solution is the position s(t) of the object at time t. 

Step-by-Step Solution

Answer

a) The differential equation:$\frac{dv}{dt}=-9.8$, $v\left(0\right)=0$

The solution is:$v\left(t\right)=-9.8t$

b)The differential equation:$\frac{ds}{dt}=-9.8t$,$s\left(0\right)=100$

The solution is $s\left(t\right)=-4.9t^2+100$

Step 1. Given information

The ball is dropped from height of 100 m.

The acceleration due to gravity is $9.8m/s^2$

Part(a). Step 1. Set up and solve a first-order initial-value problem whose solution is the velocity v ( t ) of the object at time t .

The differential equation is:

$\frac{dv}{dt}=-9.8$,  $v\left(0\right)=0$

$$\begin{gathered} \frac{dv}{dt}=-9.8 \\ dv=-9.8dt \\ \int dv=-\int 9.8dt \\ v=-9.8t+c \end{gathered}$$

When $v\left(0\right)=0$

Then

$$\begin{gathered} 0=-9.8\left(0\right)+c \\ c=0 \end{gathered}$$

So the solution to the differential equation is:

$v\left(t\right)=-9.8t$

The acceleration has a negative sign since the object falls downward.

Part(b). Step 1. To set up and solve a first-order initial-value problem whose solution is the position s(t) of the object at time t by using the answer of part (a).

The differential equation of position:

 $$\begin{gathered} \frac{ds}{dt}=v \\ \frac{ds}{dt}=-9.8t \end{gathered}$$

The initial height is 100 m.

So, $s\left(0\right)=100$

Now solve the differential equation:

$$\begin{gathered} \frac{ds}{dt}=9.8t \\ ds=-9.8tdt \\ \int ds=\int -9.8tdt \\ s=-9.8\left(\frac{t^2}{2}\right)+c \\ s=-4.9t^2+c \end{gathered}$$

Since, $s\left(0\right)=100$

Hence, 

$$\begin{gathered} 100=4.9\left(0\right)+c \\ c=100 \end{gathered}$$

So the solution of the differential equation  is:

$s\left(t\right)=-4.9t^2+100$