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Speed and Velocity - Definitions, Formulae, Practice problems, FAQs

Speed and Velocity - Definitions, Formulae, Practice problems, FAQs

Sachin starts his car and steps on the accelerator pedal. In order to know how fast he goes, he keeps an eye on his speedometer; it revs up from 0 to 50 km/hr. Now what does this 50 km/hr mean? A simple understanding of mathematics tells us that the car covers 50 km in 1 hour. Now what if Sachin recorded the same 50 km/hr moving in the eastward direction ? Now it's no longer relevant to say that Sachin’s speed is 50 km/hr. We must talk about the direction in which he is speeding as well. In other words, speed is simply a measure of how fast or slow an object is moving. When both direction and magnitude are taken into account, we are talking about its “velocity”. Although both speed and velocity appear same, they are quite different when we put things into perspective. In this article, we will explore speed and velocity in detail.

Table of contents

  • Speed
  • Average speed
  • Instantaneous speed
  • Velocity
  • Average velocity
  • Instantaneous velocity
  • Difference between speed and velocity
  • Practice problems
  • FAQs

Speed

The ratio of the distance traveled by a body to the time taken is called its speed.

Mathematically,

Speed=Distance traveledTime taken

Example, we can say that the speed of a car is 20 m/s. It is a measure of how fast it is moving. It is a scalar quantity.

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The SI unit of speed is m/s. It has a CGS unit of cm/s and its dimensional formula is [LT-1].

Average speed

Average speed(vavg) is defined as the total distance divided by the total time taken.

vavg=Total distanceTotal time 

Instantaneous speed

The speed of a particle at a certain instant of time is known as its instantaneous speed (vinst)

Let s indicate the distance traveled in time t. Then ,

vinst=LimitΔt0(ΔsΔt )=dsdt

The tangent drawn to the s-t curve makes an angle with the +ve direction of x-axis. Then slope of the curve is given by tan .

In other words, the slope of the distance time graph gives speed.

Velocity

It is the change in position or displacement of a body divided by time. Displacement comes into the picture when we talk about the shortest distance between two points. Mathematically,

Velocity=DisplacementTime taken

Average velocity

Average velocity (vavg) is defined as the total displacement divided by the total time taken.

vavg=Total displacementTotal time 

Instantaneous velocity

The velocity of a particle at a certain instant of time is known as its instantaneous velocity.

vinst

Let ds indicate the displacement traveled in time dt.

Then ,

vinst=LimitΔt0(ΔsΔt )=dsdt 

The graphical sense of dsdt is the slope of the displacement - time graph at a given time as shown in the figure. The tangent drawn to the s-t curve makes an angle with the +ve direction of x-axis. Then slope of the curve is given by tan .

In other words, the slope of the displacement time graph gives velocity.

Difference between speed and velocity

Speed

Velocity

1) It is the total path length divided by the time taken.

It is the ratio of the displacement to the time taken.

2) It is a scalar quantity.

It is a vector quantity.

3) Speed can never be negative.

Velocity can be negative or positive.

Video explanation

https://www.youtube.com/watch?v=kaLpnbmHPVY&t=853s

Practice problems

Q. The displacement of a particle varies according to the relation x=-t+t,2 , , are +ve constants. The instantaneous velocity of the particle at time t=1 s is

(a)-+2 (b)+2 (c)2- (d)None of these

A. c

Given,

Displacement , x=-t+t2

Instantaneous velocity, v=dxdt=ddt(α-βt+γt2)=-β+2γt

At t=1 s, v=-+2.

Q. The displacement of a particle changes from time t= 1 s and t= 4 s. The positions of the particle at the respective instants of time are r1=3i-4j and r2=2i-j. The velocity of the body in (m/s) is

(a)-i^+3j^3 (b)2i^+j^3 (c)i^+j^2 (d) 2i^+j^2

A. a

Given, t=4-1=3 s.

Velocity=DisplacementTime taken=r2-r1Δt=(2i^-j^)-(3i^-4j^)3=-i^+3j^3

Q. A boy travels 10 km north and 5 km east in 5 s. Find out his velocity in (m/s). Consider i in the east direction and j in the north.

A.

Given,

r1=10j^  and r2=5i^

Time t=5 s.

Velocity=Change in position vectortime

=10j+5i^ ^5=i^+2 j.^

Q. Rajesh has to travel a total distance of 60 km. If he travels half of the total time with a speed of 80 km/hr and in the rest half of the time with a speed of 40 km/hr, the average speed of Rajesh would be

(a) 60 km/hr (b) 80 km/hr (c) 120 km/hr (d) 180 km/hr

A. a

Given, v1=80 km/hr and v2=40 km/hr

If t represents the time taken for half the journey, then total time for the entire journey would be

=2t.

Total displacement=v1t+v2t=80t+40t.

The average velocity vav=DisplacementTotal time =80t+40t Total time =120t 2t =60 km/hr.

FAQs

Q. What is the main difference between speed and velocity?
A.
Speed is a measure of how fast or slow an object moves but velocity also tells us about the direction in which the speed is directed.

Q. Can velocity be negative?
A.
Velocity is direction dependent. If we consider a particular direction as positive, then just the opposite to it will be negative. When a body moves in the negative direction, then its velocity is negative.

Q. What is the difference between rest and motion?
A.
Rest is the state in which the body does not move wrt an observer. A baby sleeping on the bed is said to be at rest wrt a person standing near the bed. On the other hand, motion is when the position of a body is constantly changing with respect to time wrt an observer. A train moving on a platform is said to be in motion wrt an observer on platform.

Q. How are velocity and distance different from each other?
A.
Distance is the total path length whereas velocity on the other hand is the total displacement upon total time taken. Distance is the quantity of the path covered without mentioning the direction. Displacement is the measure of distance between the final and initial position with direction. So it is a vector quantity. The velocity is the ratio of displacement to the time taken for the displacement.

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