What is Speed, Displacement, Velocity, Acceleration - It's Introduction, Types, Mathematical Expressions, Examples

Average and Instantaneous Speed:

Since driving or riding in cars is a common activity in our daily lives, we are familiar with the concept of speed. Most of us have had experience in reading a speedometer.

Average Speed :-

What does it mean to say that we are traveling at a speed of 55 km/h? It means that we would cover a distance of 55 kilometres in a time of 1 hour if we traveled steadily at that speed. Carefully note the structure of this description: There is a number, 55, and some units or dimensions, miles per hour. Numbers and units are both essential parts of a description of speed. The term kilometres per hour implies that kilometres are divided by hours in arriving at the speed. This is exactly how we would compute the average speed for a trip.

We define the average speed as the distance traveled divided by the time it took to do the traveling, or

                              distance traveled
Average speed = -----------------------------
                                 time elapsed

Suppose, for example, that we travel a distance of 250 km in a time of 5 hours. The average speed is then 250 km divided by 5 hours, which is equal to 50 km/h. This type of computation is familiar to most of us. But average speed isn't just the mean of differentspeeds. Ah, the classic mistake! People often think if they go 60 km/h one way and 40 km/h the other, the average is 50 km/h. But that's incorrect because time spent at each speed differs. For instance, if the distances are equal, say 60 km each way, the time to go is 1 hour (60 km at 60 km/h), but returning takes 1.5 hours (60 km at 40 km/h). Total distance is 120 km, total time 2.5 hours, so average speed is 48 km/h. That's important to highlight because it's a common confusion.

Speed is independent of the direction of motion, and like all such quantities that have nothing to do with spatial orientation such as length, temperature, time, mass, density, charge, and volume, it has only a scale or size and no associated direction. Accordingly, it’s called a scalar quantity.

Constant Speed:

Uniform speed corresponds to the traversal of equal distances in equal intervals of time of any fixed duration. All of us have traveled in automobiles and have some intuitive sense of what constant or uniform speed is just keep the speedometer locked at, say, 55 km/h and you have it. If the interval is large, a body moving alternately faster and then slower can still have the same average speed over each such interval of time and so seem to be moving with a constant speed. When the speed is constant, the distance traversed during any time period can be determined.

If Vconst is the constant speed with which some object moves, then that is clearly also its average speed,
and Eq becomes Vconst = l/t. The path length traveled in a given time at a fixed speed is:
l = Vconst × t

Instantaneous Speed :-

When scientists dealt successfully with the idea of uniform speed. But when they tried to take the next logical step, to define the speed at any moment, they failed. They thought, how does instantaneous speed differ from average speed? They lacked the mathematical imagery of motion that Newton would create centuries later the calculus of change. The instantaneous speed tells us how fast we are going at a given instant but tells us little about how long it will take to travel several miles, unless the speed is held constant. The average speed, on the other hand, allows us to compute how long a trip might take but says little about the variation in speed during the trip.

If we travel a distance of 250 kilometres in 5 hours, as in our earlier example, is it likely that the entire trip takes place at a speed of 52 km/h? Of course not; the speed goes up and down as the road goes up and down, when we overtake slower vehicles, when rest breaks occur, or when the highway patrol looms on the horizon. If we want to know how fast we are going at a given instant in time, we read the speedometer, which displays the instantaneous speed.

But how do we compute this rate? What time interval should we use? What is an instant in time? Our solution to this problem is simply to choose a very short interval of time during which a very short distance is covered and the speed does not change drastically. If we know, for example, that in 1 second a distance of 20 meters was covered, dividing 20 meters by 1 second to obtain a speed of 20 m/s would give us a good estimate of the instantaneous speed, provided that the speed did not change much during that single second. If the speed was changing rapidly, we would have to choose an even shorter interval of time. In principle, we can choose time intervals as small as we wish, but in practice it can be hard to measure such small quantities.

Distance and Displacement :-

What is difference between distance and displacement. You need to distinguish these concepts clearly. Distance is the actual path length of travel between two different times. Displacement means the change in position between two different times.
Its magnitude is the euclidian distance between these two locations. The length of the displacement vector coincides with distance if and only if the path follows a straight line. In all other cases, the distance is strictly larger than the magnitude of the displacement vector.

For example, u(t1)−u(t0) is the displacement from position at t0 to position to t1. This value has nothing to do with the actual distance you have traveled. Consider an example of walking along a straight line. Suppose, you walk 10 km East, then 5 km North, then 10 km West, then 5 km South, ending up back where you started. Thus, as far as positions are concerned, your displacement is just 0 km, since start and end positions are the same. But you have walked a total distance of 30 km.

Displacement is a vector quantity, it has both Magnitude i.e the distance of the straight line (e.g., 10 meters, 5 kilometers) and Direction i.e the line's orientation from start to end (e.g., 30° north of east, left, downward). Distance is a scalar quantity. It measures the total length of the actual path traveled by the object, regardless of its starting or ending point.

VELOCITY :-

Displacement is only a vector quantity with direction. Now velocity takes that concept further by adding the time element. To describe the motion of an object completely, we must specify both the speed and its direction. The single concept that at once embraces both speed and direction is called velocity. A situation in which several motions occur simultaneously in different directions can only be analyzed using velocities. For example, a boy running east at 5 m/s throws a ball north at 10 m/s with respect to him. In what direction and at what speed relative to the ground will the ball travel?

Velocity is a vector that describes how fast an object is moving and in what direction it is moving. Velocity has both magnitude, i.e How fast the position is changing (e.g., 50 km/h, 10 m/s) and direction, i.e the direction of the change in position (e.g., North, 30° above the horizontal, downward). e.g. 60 km/h to East describes a velocity.

If object is stationary (e.g. a parked car), the velocity is zero. If object have steady speed and unchanging direction, then velocity is constant. If object have speed and direction changes, then known as changing velocity. Now, Imagine you are driving a car around a curve and you maintain a constant speed of 60 km/h. Is your velocity also constant in this case? The answer is no, m because velocity involves the direction of motion, as well as how fast the object is going. The direction of motion is changing as the car goes around the curve.

To simply state this distinction, speed as we have defined it tells us how fast an object is moving but says nothing about the direction of the motion. Velocity includes the idea of direction. To specify a velocity, we must give both its size or magnitude (how fast) and its direction (north, south, east, up, down, or somewhere in between).

Average vs. Instantaneous Velocity:

Average Velocity :-

The total displacement divided by the total time taken for a journey. It tells you the overall rate and direction of position change for the entire trip.
e.g. A car drives 60 km North in 1 hour, then 60 km South in 1 hour. Total displacement is 0 km (since car drives back to same position). It takes 2 hours for trip. Average velocity is 0 km/h.

Instantaneous Velocity :-

The velocity of an object at a specific instant in time (as Δt approaches zero). It's the velocity shown by your car's speedometer combined with your direction of travel at that exact moment. Instantaneous Velocity measures, how fast and in what direction an object is moving right now.
e.g. A car's speedometer reads 50 km/h North at 4:00 PM. This is its instantaneous velocity at that exact time. At 4:05 PM, it might be 60 km/h East ( velocity changed).

Acceleration:

Acceleration is the rate at which an object's velocity changes over time. It describes how quickly an object speeds up, slows down, or changes direction. Acceleration is a familiar idea. We use the term in speaking of the acceleration of a car away from a stop sign or the acceleration of a running back in football. We feel the effects of acceleration on our bodies when a car’s velocity changes rapidly and even more strikingly when an elevator lurches downward, leaving our stomachs slightly behind. These are all accelerations. You can think of your stomach as an acceleration detector, a roller coaster gives it a real workout. Understanding acceleration is crucial to our study of motion. Acceleration is the rate at which velocity changes.

It is vector quantity, it has magnitude i.e. how fast velocity changes and direction i.e. how the velocity vector changes. It's unit is m/s² or km/h².

Acceleration is defined to be how fast your velocity is changing. So in terms of units, you would ask how many metres per second (m/s) do you gain every second (s), or m/s/s, which is mathematically equivalent to m/s². You don't actually have to square time to find the acceleration, you just have to figure out how your velocity is changing with time.

Think of acceleration as the ohh moment, when you feel when something starts, stops, or turns. One of the attributes they will sell you the car on, in terms of measuring its acceleration, is how many seconds it takes to reach a speed, typically 60 mph. So a car might take 10 seconds to reach 60 miles per hour. 

Acceleration is the sensation you feel when your motion is changing, when you start, stop, speed up, slow down, or turn. You don't feel constant motion; you feel changes in motion. You feel acceleration, when it pushes into the floor or that floating/lightness sensation? That's your body reacting to acceleration. Direction is crucial. Accelerating upward makes you feel heavier. Accelerating downward makes you feel lighter. The direction of the acceleration matters immensely.

When your velocity is increasing forward. You feel pushed back into your seat. Acceleration forward. When you drive at steady speed i.e constant velocity. You feel normal. No acceleration. When you press the brake pedal. You move forward. Your forward velocity is decreasing so acceleration is in the opposite direction i.e. backward. That pushed into your seat, stomach drop or lurching forward feeling? That's acceleration! It's your body telling you your velocity vector is changing.

Overall, acceleration is the change. It's not just moving fast or slow. It's the moment when your speed increases, decreases, or your direction changes.

e.g.

Suppose that your car, pointing due east, starts from a full stop at a stop sign, and its velocity increases from zero to 20 m/s. The change in velocity is found simply by subtracting the initial velocity from the final velocity (20 m/s − 0 m/s = 20 m/s). To find its rate of change, however, we also need to know the time needed to produce this change. If it took just 5 seconds for the velocity to change, the rate of change would be larger than if it took 30 seconds.

Suppose that a time of 5 seconds was required to produce this change in velocity. The rate of change in velocity could then be found by dividing the size of the change in velocity by the time required to produce that change. Thus, the size of the average acceleration, a, is found by dividing the change in velocity of 20 m/s by the time of 5 seconds, we get acceleration equal to 4 m/s².

Acceleration depends on the direction of v, not just speed. During speeding up, acceleration and velocity are in same direction. e.g., car accelerating forward. During slowing down, acceleration and velocity are in opposite direction. e.g. braking while moving forward i.e. accelerating backward.

An object can accelerate even if speed is constant e.g., turning a corner at 60 km/h.

During car accelerating, velocity goes from 0 to 30 m/s in 10 s. It is accelerating 3 m/s² forward. During braking or decelerating, velocity goes from 20 m/s to 0 m/s in 4 s. It is accelerating -5 m/s², negative because acceleration opposite to motion.

Satellite orbiting earth with constant speed, but direction changes. Acceleration toward Earth’s center.
Throwing a ball upward, velocity decreases i.e. slowing down, acceleration = -9.8 m/s². Gravity pulls downward.