Measurement of Speed

Introduction

The physical quantities which contain both magnitude and direction are called vector quantities like Displacement, Velocity, and so forth. The physical quantity which contains only magnitude is called a scalar quantity.

For example, distance, speed, etc. angular speed is a scalar quantity. Measurement is the comparison of an unknown quantity with a known value. Measurement is a system of measuring. Measurement is always a numerical value. Measurements include length, mass, volume, temperature, etc. It is associated with geometry, trigonometry, algebra, etc. Units were given to the measured values. Fundamental units for length, mass, and time are m, kg, and sec. Now, we will learn the speed, their types and solve some problems related to speed.

What is speed?

Speed is the term used to know how fast or slow the object is moving. Speed has only magnitude so it is a scalar quantity. The distance travelled by the body at a particular time is the speed of the body. if the body moves fast then its speed is high. If the body moves slowly then its speed is low. the speed of the body is zero if it is at rest. The formula for speed is

$$\mathrm{speed(v)\:=\:\frac{distance}{time}}$$

The S.I unit of speed is m/s. In the cgs system, it is denoted as cm/s. It is also denoted by km/sec, km/hr, miles/hr etc. The dimensional formula for speed is LT-1. If the body undergoes linear motion then the speed of the body is denoted as linear speed and if the body undergoes rotational motion then it is denoted as angular speed. Angular speed is calculated from the total angular distance traveled by the object for a complete rotation in a particular time.

$$\mathrm{\omega\:=\:\frac{2\pi}{t}}$$

ω - angular speed

t - time taken for one complete rotation

Types of speed

The speed gives the nature of the movement of the object. They are classified into four variable speeds, uniform speed, instantaneous speed, and average speed.

Uniform speed

Let us consider a moving object. Note the distances traveled by the object for a periodic time. If the object covers equal distances in equal time intervals then the speed of the object is called uniform speed.

Variable speed

It is controversial with uniform speed. So it is also called non-uniform speed. If the object covers unequal distances in equal time intervals then the speed of the object is called uniform speed.

Average speed

If the object moves at different speeds and covers different distances then the average speed is calculated by the ratio between the total distance traveled by the object and the total time taken to cover the distance. let us consider an object moving at different speeds 𝑣1, 𝑣2, . .. covers different distances as 𝑠1, 𝑠2, . ... The time taken for the distance

$$\mathrm{average\:speed(v)\:=\:\frac{total\:distance}{total\:time}}$$

$$\mathrm{v\:=\:\frac{s_{1}\:+\:s_{2}\:+\:......\:+\:s_{n}}{\frac{v_{1}}{s_{1}}\:+\frac{v_{2}}{s_{2}}\:+\:.......\:+\:\frac{v_{n}}{s_{n}}}}$$

$$\mathrm{v\:=\:\frac{s_{1}\:+\:s_{2}\:+\:........\:+\:s_{n}}{t_{1}\:+\:t_{2}\:+\:.....\:+\:t_{n}}}$$

Instantaneous speed

Let us consider the object that moves at different speeds. The speed of the object at any instant of the time is instantaneous.

The calculation for speed

Let’s us consider a body that moves in a linear path. For a particular time, the distance travelled by the body is S. The time taken by the body to travel the distance S is t. Then the speed of the body is the ratio between the distance travelled and the time taken to travel.

$$\mathrm{v\:=\:\frac{s}{t}}$$

The speed of light in a vacuum is 3 𝑥 108𝑚/𝑠. The linear speed of the earth around the sun is 30 km/sec. The angular speed of the earth is 7.28 × 10−5𝑟𝑎𝑑/𝑠𝑒𝑐.

Relation between angular speed and linear speed

The Angular speed is calculated in terms of linear speed.

The linear speed of the particle $\mathrm{v\:=\:\frac{ds}{dt}}$

Angular speed of the particle $\mathrm{\omega\:=\:\frac{d\theta}{dt}}$

Multiply both sides of the equation by r we get,

$$\mathrm{r\omega\:=\:r\frac{d\theta}{dt}\:{rd\theta\:=\:ds}}$$

$$\mathrm{r\omega\:=\:\frac{ds}{dt}}$$

$$\mathrm{r\omega\:=\:v}$$

The equation that relates speed and the angular speed is $\mathrm{v=\:r\omega\:}$

It can be modified to calculate the angular speed as $\mathrm{\omega\:=\:\frac{v}{r}}$

Solved examples

1. Raju is driving a car with a speed of 15km/hr for 3 hours. Find the distance travelled by him?

Given − $$\mathrm{speed(v)\:=\:15km/hr}\:\:\mathrm{time(t)\:=\:3hr}$$

$$\mathrm{speed\:of\:the\:car\:(v)\:=\:\frac{s}{t}}$$

$$\mathrm{Distance\:travelled\:by\:the\:car\:(s)\:=\:vt}$$

$$\mathrm{s\:=\:15km\:/\:hr\:\times3\:hr}$$

$$\mathrm{s\:=\:45km}$$

2. If the blade of a fan rotates at an angular speed of 16π rad/sec. How much time does it take to complete one rotation?

Ans − We know that $$\mathrm{Angular\:speed\:=\:\frac{Angular\:displacement}{time}}$$

Given − $$\mathrm{\omega\:=\:16\pi\:rad/sec}$$

$\mathrm{Angular\:displacement\:for\:one\:rotation\:is\:=\:2\pi\:rad}$

$$\mathrm{t\:=\:\frac{\theta}{\omega}}$$

$$\mathrm{t\:=\:\frac{2\Pi}{16\Pi}}$$

$$\mathrm{t\:=\:0.125\:sec}$$

Conclusion

Speed gives the measurement that how fast or slow a body is moving. In vehicles, it is measured by odometer. It is of various types like uniform, non-uniform, average, and instantaneous. It is a scalar quantity because it does not have direction. In circular motion, it is known as angular speed. In this tutorial, the facts about speed and types of speed were discussed. The calculation for speed and some solved examples were also discussed. The relation between angular and linear speed is also cover in this tutorial.

FAQs

1. Describe Newton’s first law of motion.

Newton’s first law implies that the particle does not change its state of motion or rest without applying an external force. For example, a box at a rest position is moved by a person by applying force to that box. Without giving force it cannot be moved. Similarly, the ball rolling cannot be stopped without applying force to stop the motion.

2. Angular speed is the same at all points. Explain why?

Angular speed is the rate of change of displacement per unit of time.in angular motion, the rate of change of displacement is the same. So angular speed is the same at all points.

3. How does an ice skater control their angular speed?

Ice skaters and ballet dancers change their speed while doing their performance. When they stretch out their hands and legs, it increases the moment of inertia, thereby decreasing the angular speed. Similarly when they hold their hands and legs closer to their body which reduces the moment of inertia and thereby the angular speed increases. This is how they control their speed during their performance.

4. How speed is measured?

The speed of the vehicle is measured using a speedometer. The angular speed of the machines is measured using a tachometer. The speed of wind is measured using an anemometer.

5. What is the difference between speed and velocity?

Speed	Velocity
It is a scalar	It is a vector
Ratio between distance and time	Ratio between Angular displacement and time