Acceleration and Displacement

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 Displacement

(Okay, NOW we're moving!)

There it is… at rest.

Then, it begins to move! Bam! Now, it's somewhere else! Where? Where is it?

Let's start with a reference point and an object some distance from it.

An object is 3 meters from a point… (That is to say, an object has an initial distance of 3 meters from a point.)

So, if it doesn't move, then (duh) the final distance from the reference point is the same as the initial distance…

df = di

That's easy…

Now, suppose it moves. And we'll call that motion ∆d… then

df = di  + ∆d

If we understand that ∆d is a function of the rate of motion, velocity, and how long it moves, then

∆d = vave • t

So, we can substitute…

df = di  + vave • t

Okay, so far so good. Now, if an object is accelerating, then the velocity is changing. We can find the average velocity as…

vave = (vf + vi)/2

where is vf  final velocity and vi is initial velocity.

But wait! There's more!

The final velocity can be found! 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

vf = vi + at

We should plug that into… something…

df = di  + vave • t

df = di  + (vf + vi)/2 • t

df = di  + (( vi + at) + vi)/2 • t

And then, math-magic occurs and…

df = di  + (vi t) + (1/2 at²)          <--- Parenthesis added for clarity

The above equation will become the starting point for many motion problems. It is worth committing to memory or recording in a convenient place for reference.

Generally, it will be written without the parenthesis as…

df = di  + vi t + 1/2 at²

 

______________________

Seriously! How about doing that again with colors?

 df = di  + ∆d

 

∆d = vave • t

vave = (vf + vi)/2

 

vf = vi + ∆v

∆v = at

vf = vi + at

 

df = di  + ∆d

df = di  + vave • t

df = di  + (vf + vi)/2 • t

df = di  + (( vi + at) + vi)/2 • t

And then, math-magic occurs and…

df = di  + (vi t) + (1/2 at²)          <--- Parenthesis added for clarity

____________________________

Math magic, anyone?

Physics of Motion: Deriving the Distance Equation