Derive the equation of velocity-time relation
Answers
Answer:
The relation between velocity and time is a simple one during uniformly accelerated, straight-line motion. The longer the acceleration, the greater the change in velocity. Change in velocity is directly proportional to time when acceleration is constant. If velocity increases by a certain amount in a certain time, it should increase by twice that amount in twice the time. 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 using algebra. Start from the definition of acceleration.
a = ∆v
∆t
Expand ∆v to v − v0 and condense ∆t to t.
a = v − v0
t
Then solve for v as a function of t.
v = v0 + at [1]
This is the first equation of motion. It's written like a polynomial — a constant term (v0) followed by a first order term (at). Since the highest order is 1, it's more correct to call it a linear function.
The symbol v0 [vee nought] is called the initial velocity or the velocity a time t = 0. It is often thought of as the "first velocity" but this is a rather naive way to describe it. A better definition would be to say that an initial velocity is the velocity that a moving object has when it first becomes important in a problem. Say a meteor was spotted deep in space and the problem was to determine its trajectory, then the initial velocity would likely be the velocity it had when it was first observed. But if the problem was about this same meteor burning up on reentry, then the initial velocity likely be the velocity it had when it entered Earth's atmosphere. 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 the velocity some time t after the initial velocity. It is often called the final velocity but this does not make it an object's "last velocity". Take the case of the meteor. What velocity is represented by the symbol v? If you've been paying attention, then you should have anticipated the answer. It depends. It could be the velocity the meteor has as it passes by the moon, as it enters the Earth's atmosphere, or as it strikes the Earth's surface. It could also be the meteorite's velocity as it sits in the bottom of a crater. (In that case v = 0 m/s.) Are any of these the final velocity? Who knows. Someone could extract the meteorite from its hole in the ground and drive away with it. Is this relevant? Probably not, but it depends. There's no rule for this kind of thing. You have to parse the text of a problem for physical quantities and then assign meaning to mathematical symbols.
The last part of this equation at is the change in the velocity from the initial value. Recall that a is the rate of change of velocity and that t is the time after some initial event. Rate times time is change. Given an object accelerating at 10 m/s2, after 5 s it would be moving 50 m/s faster. If it started with a velocity of 15 m/s, then its velocity after 5 s would be…
15 m/s + 50 m/s = 65 m/s
Explanation:
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