Question

An airplane is flying at a speed of \(350{mi}/{h}\;\) at an altitude of one mile and passes directly over a radar station at time t = 0.

(a) Express the horizontal distance d (in miles) that the plane has flown as a function of t.

(b) Express the distance s between the plane and the radar station as a function of d.

(c) Use composition to express s as a function of t.

Short answer

(a) The required function is,\(d(t) = 350t\).

(b)The required function is,\(s(d) = d(t) + \frac{1}{2}g{\left( {\frac{{d(t)}}{{350}}} \right)^2}\).

(c)\(s(d) = 350t + \frac{1}{2}g{t^2}\)is the required function.

Step-by-step solution

Introduction

Two functions can be expressed as a combination of each other with the addition, subtraction, multiplication, and divide operation. One or more operations can be applied at a time.

If f and g are two functions they can be written as,

\(\left(f+g\right)\left(x\right),\left(f-g\right)\left(x\right),\left(f\right)\left(x\right),\left({f}/{g}\; \right)\left(x\right)\)

Given

It is given that speed is 350mi/h. And time tis 0.

(a) Horizontal distance

The gravity for the horizontal travel will be zero, so from the horizontal motion law

\(\begin{array}{l}d(t) = ut\\d(t) = 350t\end{array}\)

Where d is horizontal distance, u is the speed, and t is the time

(b) Distance between the plane and radar   

Newton’s laws of motion show the relationship between the moving particle or object’s displacement, initial and final velocity, acceleration, and time.

According to newton’s laws of motion,

\(s = ut + \frac{1}{2}g{t^2}\)

Where,\(s\),\(u\),\(t\), and g are displacement, initial velocity, time, and acceleration respectively.

Here,s is represented as the function of d,

\(\begin{array}{l}s(d) = ut + \frac{1}{2}g{t^2}\\s(d) = 350\left( {\frac{{d(t)}}{{350}}} \right) + \frac{1}{2}g{\left( {\frac{{d(t)}}{{350}}} \right)^2}\\s(d) = d(t) + \frac{1}{2}g{\left( {\frac{{d(t)}}{{350}}} \right)^2}\end{array}\)

Here g is gravity.

(c) s as a function of t

The distance s as the function of t will be,

\(\begin{array}{l}s(d) = ut + \frac{1}{2}g{t^2}\\s(d) = 350t + \frac{1}{2}g{t^2}\end{array}\)

So, the required function is \(s(d) = 350t + \frac{1}{2}g{t^2}\).