What Is Speed?

Definition: The distance traveled by a moving object in a unit time interval is called the speed of the object. Speed is also a scalar quantity like distance, which means it has only magnitude but no direction, which refers to "how fast an object is moving.

Units: Speed is measured in meter/second (m/s) and centimeter/second (cm/s) which are the S.I and C.G.S units of it. But there are also other units of speed like kilometer/hour (Km/h) etc.

What are speed and velocity? Definitions, types, and differences

Formula: If an object travels a distance s in time t, then its

${\color{Red} Speed(v)=\frac{Distance(s)}{Time(t)}}$

Almost everyone knows that a fast-moving object has a high speed and travels a relatively large distance in a short time. Or a slow-moving object that has a low speed, covers a relatively small distance in the same amount of time.

This means speed is directly proportional to distance when time is constant i.e v ∝ s (t constant). And the speed is inversely proportional to time when distance is constant i.e v ∝ 1/t (s constant).

So combining these two rules together the formula of the speed can be written as

${\color{Red} v=\frac{s}{t}}$

So in order to measure the speed, we need to know how far the object has traveled and how long it has taken to get there. Let's understand with the help of an example. Suppose you are driving a car from your home to your office.

Let's the distance between the home and the office be 80 km and the time taken to reach the office is 2 hours. Then what is the speed of your car? Let's find out

${\color{Red} Ans:\; Speed(v)=\frac{80}{2}\;\frac{km}{h}=40\frac{km}{h}}$

So your car's speed is 40km/h.

What Are Uniform And Non-uniform Speed?

Uniform speed: An object is said to be moving with uniform speed if it covers equal distances in equal intervals of time throughout the journey.

For example, the motion of a ball on a frictionless plane surface.

Non-uniform speed: An object is said to be moving with non-uniform speed (or variable speed) if it covers unequal distances in the same intervals of time.

For example, the motion of a ball on a rough surface, the motion of a car on a crowded street.

What Is Average Speed?

Definition: The ratio of the total distance traveled by an object to the total time of journey is called the average speed of the object.

The equation above gives us the answer, but how accurate is it? Is the car speed 40 km/h? No, it is the average speed of your car. Since your car's speed during this journey must certainly have varied because of the traffic flows or the other activities that were happening on the road.

So to emphasize this point, the equation of the speed is sometimes modified as follows

${\color{Red} Average \;speed\;(\bar{v})=\frac{Total \;distance\; travel}{Total\;time\;taken}}$

This is not the final idea of speed. So you need to know another concept of speed called instantaneous speed.

What Is Instantaneous Speed?

Definition: If the speed of a body keeps on changing continuously with time, its speed at any instant is said to be as instantaneous speed.

It can be measured by measuring the distance traveled by the body in a very short time interval and then dividing the distance with the time interval. In the case of a vehicle, the speedometer of that vehicle measures the instantaneous speed.

So to measure the instantaneous speed of a body we take a very short time interval Δt in which the body travels a distance Δs. So the formula goes like this

${\color{Red} \Delta v=\frac{\Delta s}{\Delta t}}$

Now the instantaneous speed is given by the limiting value of the average velocity as the time interval approaches zero. Learn more about Limits (mathematics).

${\color{Red} v=\lim_{\Delta t\rightarrow 0}\frac{\Delta s}{\Delta t}=\frac{ds}{dt}}$

In the case of position vector s is replaced by r. Here the position vector r is used to represent the two and three-dimensional motion of the object.

${\color{Red} v=\lim_{\Delta t\rightarrow 0}\frac{\Delta r}{\Delta t}=\frac{dr}{dt}}$

In three dimensions the position vector is represented as

${\color{Red} r=x\hat{i}+y\hat{j}+z\hat{k}}$

Where x, y, and z are the components in a three-dimensional (3-D) coordinate system, where i, j, and k are the unit vector that are showing particular directions. So we can express v in terms of component form and it is represented as

${\color{Red} v=\frac{dr}{dt}=\frac{d}{dt}(x\hat{i}+y\hat{j}+z\hat{k})}$

${\color{Red} Or\;v=\hat{i}\frac{dx}{dt}+\hat{j}\frac{dy}{dt}+\hat{k}\frac{dz}{dt}}$

${\color{Red} Or\;v=v_{x}\hat{i}+v_{y}\hat{j}+v_{z}\hat{k}}$

Here, vx = dx/dt, vy = dy/dt, and vz = dz/dt show the velocty component in a three-dimensional (3-D) coordinate system.

What Is Velocity?

Definition: The distance traveled by a moving object in a specified direction in a unit time interval is called the velocity of the object. Velocity is also a vector quantity like displacement, which means it has both magnitude and direction.

So when evaluating the velocity of an object It is necessary to keep tracking of the direction i.e one must include directional information in order to fully describe the velocity of the object. For example, if a car moves towards the west with a speed of 40 km/h then its velocity is 40 km/h, west.

In many cases, speed and velocity are the same when we solve different problems in physics but it is notable that the speed has no direction (it is a scalar quantity) and velocity is the value of speed including direction.

Formula:

Velocity is a change in objects displacement (s) over time (t). So

${\color{Red} Velocity(v)=\frac{Displacement(s)}{Time(t)}}$

Where Displacement (s) = final position (x2) – initial position or change in position (x1)

Time (t) = the time in which the change occurs (s).

Difference Between Speed And Velocity

Just as distance and displacement have a distinct meaning in spite of their similarities, similarly speed and velocity also have a distinct meaning. Below is a brief description of the differences between speed and velocity

Speed

Velocity

1. The distance traveled by a moving object in a unit time interval is called the speed of the object.

1. The distance traveled by a moving object in a specified direction in a unit time interval is called the velocity of the object.

2. Speed is a scalar quantity, it does not indicate the direction of motion.

2. Velocity is a vector quantity. It tells us the speed as well as the direction of motion.

3. The speed is always positive

3. The velocity can be positive or negative depending on the direction of motion.

4. In one round in a circular path, the average speed of the object is not zero.

4. In one round in a circular path, the average velocity of the object is zero.

Applications Of v = ds/dt:

Example: The position of a particle as a function of time is given as s(t) = (t^3+8t^2-t)m. Find the instantaneous speed of the particle at t = 2s?

Answer:

The instantaneous speed is given by v = ds/dt.

$Since\; s(t)=t^{3}+8t^{2}-t$

$v=\frac{ds}{dt}=3t^{2}+16t-1$

$At\;t=2s,\;\left [ \frac{ds}{dt} \right ]_{t=2}=3.2^{2}+16.2-1=43$

$\therefore \;\left [ v \right ]_{t=2}=43\;m/s$

Ans: the speed of the particle at t = 2s is 43 m/s.

Example: The position of a particle is defined by s(t) = (4t^3 – 3t^2 + 2t)m. At what time ( time in seconds) is its instantaneous speed equal to 384? Do it yourself

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What are speed and velocity? Definitions, types, and differences