Chapter 3, Problem 3.1CYU

Check your Understanding A cyclist rides 3 km west and then tune around and tides 2 kin east. (a) What Is his displacement? (b) What is the distance traveled? (c) What is the magnitude of his displacement?

To determine

(a)

The displacement of the cyclist.

Answer to Problem 3.1CYU

The displacement of the cyclist is $-1\text{km}$.

Explanation of Solution

Given:

Initially cyclist travel towards west and covers 3 km and then he turned towards east and travelled covering 2 km.

Formula used:

Displacement is defined as the change in position of an object. Numerically it is written as,

$\Delta x=x_f-x_i$

Where,

$x_f$ is the final position of the body.

$x_i$ is the initial position of the body.

Also, when there is a series of displacement, then total displacement is equal to the sum of individual displacements and it is written as,

$\Delta x_T={\displaystyle \sum \Delta x_i}$

Calculation:

Consider that the cyclist starts from origin and east direction will be positive.

Initially cyclist moves 3 km towards west. The initial position of cyclist is 0 km and final position of cyclist is $-3\text{km}$. So, the displacement for the first case will be,

$\begin{array}{c}\Delta x_1=\left(-3\right)-0\\ =-3\text{km}\end{array}$

After travelling 3 km towards west, cyclist turns around and travels 2 km towards east. Now, the initial position of cyclist is $-3\text{km}$ and final position is $-1\text{km}$. So, the displacement for the second case will be,

$\begin{array}{c}\Delta x_1=\left(-1\right)-\left(-3\right)\\ =-1+3\\ =2\text{km}\end{array}$

The total displacement will be,

$\begin{array}{c}\Delta x_T=\Delta x_1+\Delta x_2\\ =-3+2\\ =-1\text{km}\end{array}$

Conclusion:

Therefore, the displacement of the cyclist is $-1\text{km}$.

To determine

(b)

The distance travelled by the cyclist.

Answer to Problem 3.1CYU

The distance travelled by the cyclist is $5\text{km}$.

Explanation of Solution

Given:

Initially cyclist travel towards west and covers 3 km and then he turned towards east and travelled covering 2 km.

Formula used:

Distance travelled by an object is equal to the sum of magnitudes of individual displacements, $\Delta x_{TD}={\displaystyle \sum \left|\Delta x_i\right|}$

Calculation:

The displacement for the first case is,

$\Delta x_1=-3\text{km}$

The displacement for the second case is,

$\Delta x_2=2\text{km}$

The total distance travelled is,

$\begin{array}{c}\Delta x_{TD}=\left|\Delta x_1\right|+\left|\Delta x_2\right|\\ =\left|-3\right|+\left|2\right|\\ =3+2\\ =5\text{km}\end{array}$

Conclusion:

Therefore, the distance travelled by the cyclist is $5\text{km}$.

To determine

(c)

The magnitude of the displacement.

Answer to Problem 3.1CYU

The magnitude of displacement of the cyclist is $1\text{km}$.

Explanation of Solution

Given:

Initially cyclist travel towards west and covers 3 km and then he turned towards east and travelled covering 2 km.

Formula used:

Displacement is defined as the change in position of an object. Numerically it is written as,

$\Delta x=x_f-x_i$

Where,

$x_f$ is the final position of the body.

$x_i$ is the initial position of the body.

Also, when there is a series of displacement, then total displacement is equal to the sum of individual displacements and it is written as,

$\Delta x_T={\displaystyle \sum \Delta x_i}$

Calculation:

The displacement for the first case is,

$\Delta x_1=-3\text{km}$

The displacement for the second case is,

$\Delta x_2=2\text{km}$

The total displacement will be,

$\begin{array}{c}\Delta x_T=\Delta x_1+\Delta x_2\\ =-3+2\\ =-1\text{km}\end{array}$

Magnitude of displacement is,

$\left|\Delta x_T\right|=\left|-1\right|=1\text{km}$

Conclusion:

Therefore, the magnitude of displacement of the cyclist is $1\text{km}$.