Q. 56 - Calculus

Question

Velocity $v\left(t\right)$ is the derivative of position $s\left(t\right)$. It is also true that acceleration $a\left(t\right)$ (the rate of change of velocity) is the derivative of velocity. If a race car’s position in miles t hours after the start of a race is given by the function $s\left(t\right)$, what are the units of $s(1.2)$? What are the units and real-world interpretation of $v(1.2)$? What are the units and real-world interpretations of $a(1.2)$? 

Step-by-Step Solution

Answer

Ans:  

Since the distance is measured in miles, therefore the unit of $s(1.2)$ will be miles.

Again derivative of position function over time gives the velocity, therefore $v(1.2)$ will represent the velocity with unit miles per hour.

Similarly, the derivative of the velocity with respect to time gives the acceleration. Therefore the unit of acceleration is miles per hour per hour.

Step 1. Given information.

given,   

    Distance function  $s\left(t\right)$

    Velocity function  $v\left(t\right)$

    acceleration function $a\left(t\right)$

Step 2. The objective is to explain the units of distance, velocity, and acceleration.

Here the function $s(1.2)$ indicates the distance covered by the object at $t=1.2$  hour. Since the distance is measured in miles, therefore the unit of $s(1.2)$ will be miles.

And the derivative of position function over time gives the velocity, therefore $v(1.2)$ will represent the velocity with unit miles per hour.

Similarly, the derivative of the velocity with respect to time gives the acceleration. Therefore the unit of acceleration is miles per hour per hour.