Velocity

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What is Velocity?

Velocity is defined as the rate and direction of the change in the position of an object or the rate of change of displacement. In calculus terms, velocity is the first derivative of position with respect to time.
The most common way to calculate the constant velocity of an object moving in a straight line is the formula: $$v = {s \over t}$$ where $v$ is the velocity $s$ is the displacement $t$ is the time it takes to complete the movement

Average Velocity

This is the first major formula, used for the calculation of average velocity. $$ \vec{v} = {\vec{Δd} \over Δt}$$ v = velocity (m/s) d = displacement (m) t = time (s)
  • It is called average velocity because it looks at your overall velocity for the entire trip, not at any one particular velocity you might have been traveling at during the trip.
  • What you have to measure is the total displacement divided by the total time.
  • If you drive 275 km to Calgary in 3.00 hours, I calculate your velocity based on this information. The velocity I calculate, 91.7km/h, will be your average velocity.
  • I'm looking at the entire trip. It would certainly be rare if you had driven at exactly 91.7km/h every single moment during your drive to Calgary.
  • Sometimes you would have been going faster, sometimes slower, but overall your average velocity was 91.7km/h.
  • You basically have to look at the start and finish only, how far did you move from where you started (displacement) in a certain amount of total time.
Example 1: I try to run the $100m$ race to break the world's record. Unfortunately, it takes me $16.83s$ to complete the run, so I don't think I'll be in the record books. Determine my average velocity.
Solution: This is an easy calculation... nothing fancy.
$$v={d \over t} = {100 \over 16.83}=5.94=5.94m/s$$
This is my average velocity. It does not show that I have to speed up at the start of the race, or that maybe I was slowing down near the end.

Speed vs Velocity

Speed is a scalar quantity, it does not account for direction, while velocity is a vector quantity which is aware of direction. If ran across the room and then returned to your original position, you would have a speed - the distance divided by the time. But your velocity would be zero since your position didn't change between the beginning and the end of the interval. There was no displacement seen at the end of the time period. You would have an instantaneous velocity if it were taken at a point where you had moved from your original position. If you go two steps forward and one step back, your speed isn't affected, but your velocity would be.
Fig.no.1: Speed vs Velocity
Here speed is the total distance covered (220m) in a specific time. While velocity can be calculated by observing the displacement (130m) in the same time interval.

Relating Velocity, Speed, and Acceleration

Speed, velocity, and acceleration are all related to each other. Remember:
Speed is a scalar quantity that indicates the rate of motion distance per time. Its units are length/time.
Velocity is a vector quantity that indicates distance per time and direction. Like speed, its units are length/time, but direction is also cited.
Acceleration is a vector quantity that indicates the rate of change of velocity. It has dimensions of length/time.

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