Chapter 2.6, Problem 18AYU

18. Uniform Motion Two cars leave an intersection at the same time. One is headed south at a constant speed of 30 miles per hour, and the other is headed west at a constant speed of 40 miles per hour (see the figure). Build a model that expresses the distance d between the cars as a function of the time t.

[Hint: At $t\text{ = 0}$, the cars leave the intersection.]

To determine

To find: To build a model that expresses distance $d$ between two cars as a function of time.

Answer to Problem 18AYU

Solution:

$d\left(t\right)=50t$

Explanation of Solution

Given:

Two cars are travelling perpendicular to each other with velocities 30 miles/hr and 40 miles/hr.

Calculation:

The motion of the cars can be represented as the figure below:

Let, $v_1$ be the velocity of the car travelling south.

Let, $v_2$ be the velocity of the car travelling west.

Let, $d_1$ be the distance at time $t$ of the car travelling south.

Let, $d_2$ be the distance at time $t$ of the car travelling west.

$d_1=v_1.t$

$d_2=v_2.t$

The distance between the two cars at any time ‘$t$’ can be written as

$d\left(t\right)=\sqrt{d_1^2+d_2^2}$

$d\left(t\right)=\sqrt{v_1^2{t}^2+v_2^2{t}^2}$

$=\sqrt{v_1^2+v_2^{2}t}$

$=\sqrt{{30}^2+{40}^{2}t}$

$=50t$

Therefore the distance between the cars at any time $t$ is $d\left(t\right)=50t$.