Wednesday, September 1, 2021

Acceleration and Final Velocity

Physics Index

Where are we going with this? The information on this page relates to the skills needed to investigate and evaluate the graphical and mathematical relationship (using either manual graphing or computers) of one-dimensional kinematic parameters (distance, displacement, speed, velocity, acceleration) with respect to an object's position, direction of motion, and time.

Acceleration and Final Velocity
(We just keep going on, don't we?)

We generally have a good idea what acceleration is, in terms of pulling away from a red light when it turns green. The vehicle moves faster. If we mash the gas pedal harder, we speed up more quickly.

Those who are into cars will discuss that one car has better acceleration than some other car.

It seems like acceleration is something we need to understand!

Acceleration is the rate at which velocity changes. Pulling away from a red light, the velocity changes from zero to something. If we are driving at 30 mph and press the gas, the velocity changes from 30 to something else!

PAUSE: think about slowing down. Slowing down is also a change in velocity. Velocity decreases when the direction of acceleration is opposite the direction of motion.

At this point, we need to drive home the concept of vectors.

A vector has both a magnitude and a direction:

• displacement
• velocity
• acceleration

Scalars have only a magnitude:

• distance traveled / distance moved
• speed
• hmmm… we can also discuss the rate at which speed changes, and that would probably be called acceleration, also, though it would be an inaccurate use of the word.

So, back to that slowing down idea…

Slowing down is a change in velocity. It can be conceived as deceleration or as a negative acceleration.

Okay, let's move on to the good stuff. The math.

Logically, you can think that the amount that velocity changes depends on…
• the rate it changes
• and how long it changes

…and that makes perfect sense. And, it is correct.

So, if v is instantaneous velocity,  and if a is rate of acceleration (the rate that velocity changes) and if t is elapsed time, then…

∆v = a • t

Some sources will call the the v-a-t formula or the vat formula.

So, what that means is this: the change in velocity is equal to the rate of change multiplied by the amount of time it is changing. Or…

the change in velocity is the rate of acceleration times the elapsed time.

Finding Final Velocity

The final velocity an object has falls under logic like this:

How fast was it going to begin with and how much did it change? In other words, what was its initial velocity and what was the change in velocity?

Or…

where vf  is final velocity and vi  is initial velocity and ∆v is the change in velocity then

vf = vi + ∆v

and since

∆v = at

then

v= vat

Naturally, if v is zero, this is easier!