Question

The net displacement of a student that goes to school from home, which is 2 km away when he travels by road, is √2 km. If he travels by car with a constant speed of 20 km/h, find the average speed and average velocity of the car.
A) ${\text{avg}}{\text{. speed}} = 20{\text{km/h}}$ and ${\text{avg}}{\text{. vel}} = 20{\text{km/h}}$
B) ${\text{avg}}{\text{. speed}} = 20{\text{km/h}}$ and ${\text{avg}}{\text{. vel}} = 10\sqrt 2 {\text{km/h}}$
C) ${\text{avg}}{\text{. speed}} = 20{\text{km/h}}$ and ${\text{avg}}{\text{. vel}} = 20\sqrt 2 {\text{km/h}}$
D) ${\text{avg}}{\text{. speed}} = 10\sqrt 2 {\text{km/h}}$ and ${\text{avg}}{\text{. vel}} = 20{\text{km/h}}$

Answer 1

Hint: The displacement corresponds to average velocity whereas the distance corresponds to average speed. The car travels with constant speed. So average speed will be the same as the constant speed. The average speed is generally equal to or greater than the average velocity.

Formulas used:
The average speed is given by, $\overline u = \dfrac{D}{t}$ where $D$ is the total length of the path or distance travelled and $t$ is the time interval in which the motion takes place.
The average velocity is given by, $\overline v = \dfrac{d}{t}$ where $d$ is the change in position or displacement and $t$ is the time interval in which the motion takes place.

Complete step by step answer:
Step 1: List the given data.
The distance between the school and home is $D = 2{\text{km}}$ and the net displacement is $d = \sqrt 2 {\text{km}}$.
The constant speed of the car is 20km/h.

Step 2: Calculate the average speed of the car.
Given, the car travels with a constant speed of 20 km/h i.e., the speed remains the same throughout the motion. Hence, the average speed will be equal to the constant speed.
Therefore, the average speed of the car is $\overline u = 20{\text{km/h}}$ .

Step 3: Calculate the average velocity of the car.
The average velocity of is given by, $\overline v = \dfrac{d}{t}$ ------ (1)
where $d = \sqrt 2 {\text{km}}$ is the displacement of the car and $t$ is the time interval and it is unknown.
To find the time interval in which the motion happens we use the relation for the average speed.
We have the average speed $\overline u = \dfrac{D}{t}$ or, $t = \dfrac{D}{{\left( {\overline u } \right)}}$------- (2)
Substituting the value for distance $D = 2{\text{km}}$ and $\overline u = 20{\text{km/h}}$ in equation (2) we get, $t = \dfrac{2}{{20}} = \dfrac{1}{{10}}{\text{h}}$
Substituting the value for displacement $d = \sqrt 2 {\text{km}}$ and $t = \dfrac{1}{{10}}{\text{h}}$ in equation (1) we get, $\overline v = \dfrac{{\sqrt 2 }}{{\left( {\dfrac{1}{{10}}} \right)}} = 10\sqrt 2 {\text{km/h}}$

Therefore the average velocity of the car is $\overline v = 10\sqrt 2 {\text{km/h}}$.

Note:
When an object undergoes a rectilinear motion, the distance and displacement of the object may differ and thereby causing the average speed and average velocity to differ too. But the time interval in which the motion takes place will remain the same.