Distance and Displacement:

Distance is the measurement of the paths taken by the body to travel from one point to another. It may or may not be the direct path from initial point to the final point. It is a scalar quantity that has magnitude only. Mathematically it is represented as:

Distance = sum of the measurement of all the paths travelled to reach from one point to another.

\[{\text{Speed}} = \frac{\text{Distance}}{\text{Time}}\].

Therefore, Distance = Speed x Time.

Displacement is the measurement of the length of the shortest path from the initial point to the final point. It is a vector quantity having direction as well as magnitude.

\[{\text{Velocity}} = \frac{\text{Displacement}}{\text{Time}}\].

Therefore, Displacement = Velocity x Time.

For example, a man has to go to the office. So he starts from his home to the office

Example: In this the total path travelled is 5 + 4 + 3 = 12 m

And 12 m is the distance.

Difference Between Distance and Displacement

Distance is a scalar quantity whereas displacement is a vector quantity.

Distance is the total path covered by the body and displacement is the shortest path covered by the body.

The distance can never be zero but displacement can be.

Distance can never be negative and displacement can be both i.e., positive or negative.

Distance doesn’t decrease with time whereas displacement decreases.

Distance is always equal to or greater than the displacement.

Solved Problems

Example 1: A body is moving in a straight line. The distances from the origin are shown with time in the given figure. A, B, C, D & E represent different motion.

Find the displacement in 4 sec.

Find the total distance travelled.

Find the displacement in the last 2 seconds.

Solution:

The displacement in 4 sec = 40 m

Total distance travelled by the body is

80 + 80 = 160m

Because the maximum distance travelled by the body is 80m and then it comes back to its initial position.

Displacement in the last 2 sec = -80m

Example 2: Closed path is travelled by a body. Point A, B, C & D represents the path. Find the displacement and total distance travelled by the body.

Solution: Distance travelled by the body is

Distance = 2πr

Where r= radius of the body = 3km

Distance = 2 π 3 = 6π km

Displacement of the body is Zero because the body started from point A and came back to its initial position A.

Example 3: Find the displacement and distance travelled by the body if the body moves from point A to B to C to D.

Solution:

Distance travelled is

= ¾ (2πr)

= ¾ (2π3)

= 4.5π km

Since ¾ is the portion travelled by the body so that’s why distance travelled is 4.5π km

Displacement -

By applying Pythagoras Theorem on the given figure we get,

AD2 = AO2 + OD2

=32+32= 18

= \[AD = 3\sqrt{2}km\]

Try Yourself:

A man starts to travel in the east direction. After travelling 20 km he turns 900 to right and then travels 4 km. Find the total distance he travelled and now he is in which direction.

A man driving his car starts from his home which is in the west direction. After travelling 10 km he took a right turn and then travel 24 km to find the total distance he travelled and now he is in which direction.

The speed of the bicycle is 10 km/hr. In 10 mins, what distance he will cover?

A sailor sails his boat. After travelling 20 km in the south direction he finds he is in the wrong direction so he moves back 5km and turns 900 clockwise and then travels 14 km. Find the total distance he travelled and the displacement.

A man starts to travel in the North direction. After travelling 20 km he turns 900 to right and then travels 4 km. Find the total distance he travelled and now he is in which direction.

A truck driver started his journey to Haryana at noon having speed 50 km/hr and reached Delhi at 6:30 pm. What is the total distance he travelled?

In a race, the path is circular of radii 7 km. All are at the starting line. As the PTI whistles, everyone started running and reached the finishing point that is one has to complete a circle. What are the displacement and distance travelled by the runners?

A snail starts to crawl in the east direction. After travelling 20 cm turns to the right and then travel 4 cm as it reached the end it again turn anticlockwise and crawl 22cm. Find the total distance he travelled and now it is in which direction.

The initial position of a car is 0 km. Lisa drove it to the west 30 km and then waited for 15 mins. Then she came back home. What is the total distance she travelled? Also, find the displacement.

Lion to trap his food just run at a speed of 20 km/hr and within 30 mins he is over his prey. How far is the prey from him?

A man starts to travel in the east direction. After travelling 33 km he turns 900 to left and then travels 4 km again he took a turn to West and travels 3 km then comes back to its initial place. What is the distance covered and the displacement?

Additional Information:

Similarities Between Distance and Displacement:

A reference point is required for the measurement of distance and displacement.

SI Unit of distance and displacement is the same that is meter (m).

If the motion of the body is in the same direction and on the straight line then both of them will be equal.

Dimension is the same in both the cases of distance and displacement.

Formula Used to Convert Inches into Centimeter-

1 Inch = 2.54 centimeters

1 centimeter = \[\frac{1}{2.54}\] inches

1 centimeter = 0.3937 inches (approximately).