Velocity-Time
The relation between velocity and time is a simple one during constantly accelerated, straight-line motion. Constant acceleration implies a uniform rate of change in the velocity. The longer the acceleration, the greater the change in velocity. If after a time velocity increases by a certain amount, after twice that time it should increase by twice that amount. Change in velocity is directly proportional to time when acceleration is constant. If an object already started with a certain velocity, then its new velocity would be the old velocity plus this change. You ought to be able to see the equation in your mind’s eye already. This is the easiest of the three equations to derive formally. Start from the definition of acceleration, expand the Δv term, and solve for v as a function of t.
Since acceleration is also the first derivative of velocity with respect to time, this equation can also be derived using calculus. Just reverse the action of the definition. Instead of differentiating velocity to find acceleration, integrate acceleration to find velocity. Since acceleration is assumed constant, this is quite easy.
The symbol v0 (v enought) is called the initial velocity. It is also given the symbol u in many texts. The initial velocity is the velocity that a moving object has when it first becomes important in a problem. For example, say a meteor was spotted deep in space and the problem was to determine its trajectory, then the initial velocity would be the velocity at the time it was observed. But if the problem were to determine its velocity on impact, then it’s initial velocity would more likely be the velocity it had when it entered the earth’s atmosphere. In this case, the answer to, “What’s the initial velocity?” is “It depends”. This turns out to be the answer to a lot of questions.
The symbol v is then the velocity some time Δt after the initial velocity is often called the final velocity. What is taken to be the final velocity is depends on the problem you are solving. There is no hard and fast rule.
The last part of this equation aΔt is the change in the velocity from the initial value. Recall that a is the rate of change of velocity and that Δt is the time interval since the object had its initial velocity v0. Rate multiplied by time equals change. Thus if an object were accelerating at 10 ms-2, after 5 s it would be moving 50 ms-1 faster than it was initially. If it started with a velocity of 15 ms-1, its velocity after 5 s of acceleration would be 65 ms-1.