# What is the difference between non-uniform velocity and acceleration?

So I know non-uniform velocity as, "Non-uniform velocity is when an object covers unequal distance in equal interval of time in a particular direction or covers equal distance in equal interval of time but changes its direction" (am I correct?) and acceleration is "rate of change of velocity per unit time". So my question is are they same or different things and if different then how? In both scenario either direction changes or magnitude or sometimes both. So how they are different?

P.S: 1) Sorry If that is a dumb question but I was teaching my little brother and it just came randomly in my mind since then it is bothering me. 2) Excuse my English. (Not a native speaker.)

If "non-uniform" velocity means that the velocity is changing, then in order for the velocity to be "non-uniform", an acceleration must take place. This is how these two concepts relate.

However, don't get tripped up by the use of the word "uniform". There exists, for example, "uniform circular motion" where the speed remains constant, but the velocity and acceleration vectors are indeed changing over time.

Non-uniform velocity as the name suggests, is still referring to velocity. Whereas acceleration is the rate of change of velocity. Just like how velocity quantifies how the displacement is changing with time, acceleration quantifies how the velocity is changing with time.

Graphically, non-uniform velocity is just the curve in velocity-time graph. And acceleration is the slope at (around) each point in the curve. As you can see, for the uniform velocity the slope (acceleration) is zero. But it is non zero for the non-uniform case. It will always be so (think about it).

Acceleration is, by definition, the change in the velocity of the object. So if acceleration is zero, then the velocity will also not change, ie. it is moving uniformly.

Barring some vague "wordplay" type of trap questions, this is actually quite a concrete definition, and it is easier to understand this by stepping time one unit at a time, looking at how the position, velocity and accelaration values change for an object.

Suppose that at t=0 (in seconds or any other unit),

Position = [0, 0, 0] Velocity = [10, 20, 30] Acceleration = [1, 0, 0]

Then, at t=1, if the acceleration is constant, the object's motion will be fully described by:

Position = [10, 20, 30] Velocity = [11, 20, 30] Acceleration = [1, 0, 0]

At t=2,

Position = [21, 40, 60] Velocity = [12, 20, 30] Acceleration = [1, 0, 0]

At t=3,

Position = [33, 60, 80] Velocity = [13, 20, 30] Acceleration = [1, 0, 0]

So, if the acceleration value is [0,0,0], then clearly the velocity will never change from the original value no matter how long time has lapsed, so the object must be moving with uniform velocity. Hope it helps!