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Speed and Velocity: Basic Difference and Formulas to Calculate Them

Speed Formula
The formula for speed is perhaps one of the most known and popular scientific formulas, mainly because its application is so widespread and prominent in our daily lives. Here is a look into this significant scientific instrument.
Kundan Pandey
Last Updated: Feb 21, 2018
Did You Know?
The noise that you hear when someone cracks a whip is because the tip of the whip moves so fast that it breaks the speed of sound.

In layman's terms, speed is usually associated with something fast. There is an assumption that if the term speed is used, it refers to something that is going or moving fast. This is not true. Speed implies the rate of rapidity of movement. If that rate is slow, the speed is slow and if that rate is fast, the speed of the object is fast.
Speed alone has no meaning or practical use. It needs many other factors like time, distance, displacement, and velocity, which are some of the fundamental physical quantities that govern the motion of an object. Understanding these fundamentals of physics is necessary to build a strong base in physics and to get an idea about how things in the physical world work.
What is Speed?

Speed is scientifically defined as the distance traveled divided by the time taken to travel that distance. Even though this might represent only one definition among hundreds of others, it is the most basic one.
Though scientific definitions of speed might have originated and evolved only a few centuries ago, the application of the concept of speed has been in existence before recorded history, simply because it is such an obvious phenomenon. Literally anything and everything that we do can be defined in terms of speed. It is believed that many ancient armies have used the concept of speed to determine how long it would take them to travel to the enemy's camp.
Speed vs. Velocity

Before we go on to the formula of speed, a factor that is of paramount importance must be considered while defining speed. A very closely related physical quantity called velocity is often confused with speed. So how are speed and velocity different?
Students of physics must be aware of scalar and vector quantities. Quantities that have specific direction and magnitude are known as vector quantities, while those representing only magnitude are called scalar quantities. Speed is a scalar quantity, while velocity is a vector quantity. Speed represents the magnitude of velocity. When you state the speed of an object, you can say, for instance 55 km/hr. However, when you state the velocity, you have to say that it is 55 km/hr east or any other direction. Direction is an important parameter for velocity. The direction of velocity is the direction in which the object moves and it is represented by a vector sign.
Formula for Speed

Now that we have discussed the definition of speed and how it differs from velocity, let's now discuss the formula for it. The speed of an object that travels a distance (d) over a period of time (t) is given by:

Speed (v) = Total Distance Traveled / Total Time Taken
v = d/t

Students with basic knowledge of calculus can also understand the instantaneous and average formula for speed. Instantaneous speed is the speed at a particular moment of time. The formula for instantaneous speed is as follows:

Instantaneous speed = dS/dT
where S is the distance covered and T is the time taken
Formula in Application

Let us take a real life example to understand how the formula for speed works. Suppose on Monday, you decide to walk to school instead of taking the bus. It takes you 30 minutes to walk to your school, which is 3 miles away from your house. Now on Tuesday, you decide that walking to the school was a tiresome process, so you decide to take the bus. Sure enough, you reach the school faster and are less tired. Let us assume it took you 5 minutes to reach your school when traveling by the school bus.
Why do you think there is such a vast amount of time difference, even though you are traversing the same distance on both days. It is because your speed in these two events is different. In the first case where you walked to school, your speed would be calculated by dividing the distance from your house to your school, that is, 3 miles, by the time taken to cover this distance, that is 30 minutes. So the speed would be 3 miles / 30 minutes or 3 miles / 0.5 hours, which is equal to 6 miles per hour or 6 mph (Note: We convert minutes to hours as the standard convention is to write the speed as miles per hour or meters per second). Using the same method, the speed in the second case is 3 miles divided by 5 minutes or 1/12th of an hour, which is equal to 36 miles per hour or 36 mph. Quite clearly, the speed in the second event, where the bus was taken, is considerably more than the speed of walking. Hence the distance was covered in a shorter time.
Formulas When Speed is Not Constant

In practical situations, speed will be variable and will not remain constant throughout. When the speed fluctuates, that phase of speed has an initial velocity and a final velocity, and the rate at which this change takes place is called acceleration. When this is the case, the following equations are used:

v = u + at
v2 = u2 + 2aS
S = ut + ½at2

where u is the initial speed, v is the final speed, a is the acceleration, S is the distance covered, and t is the time to taken to cover the distance
Angular Speed Formula

As one learns more about motion equations and mechanics, they will also gain knowledge about rotatory motion. Just like linear speed is concerned with one dimensional motion, angular speed is related to circular motion. The formula for angular speed is given by:

v = ωr
where v is linear speed, ω is angular speed and r is the radius of the circle.

This can be applied to any circular movement of an object, for example, the Earth spinning around the Sun, the circular motion of a wheel of a car, a racing car going around a circular track, etc.
So now that you know about how the formula of speed works, you can use it not only for solving problems but also apply the concept in real life situations to gain a better understanding of the physical world around us.
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