Equations of Motions

Problems on bodies moving with CONSTANT acceleration can be solved quickly using the equation of motion. There are 4 equations:

1. v = u + at
2. s = 0.5(u+v)t
3. s = ut + 0.5 at2
4. v2 - u2 = 2as

where [u = initial velocity] [a = acceleration] [v = final velocity] [s = displacement] [t = time taken]

If we know any 3 of s, u, v, a and t, then others can be found from the equations by substitution.

Why only for constant acceleration?

But, why can only question with uniform acceleration can be solved with these equation? Where are these equations come from? What to do if the acceleration is not constant? Here comes the answer!!!

Integrate the above equation with respect to t, we get

Actually C1 represents the initial value, we put it as initial velocity u. Then we get

v = u + at (equation 1)

Integrate equation 1 with respect to t again, you will get

s = ut + 0.5at2 (equation 3)

There should be one more constant some out from equation 3, but as we assume the initial position is 0, so the constant is 0 too.

From equation 1 and 3, if you eliminate a, you will get equation 2. If you eliminate t, you will get equation 4. Oh, it is so easy to get these 4 equations. I think you now also understood why these equation must work on constant acceleration.

What to do if the acceleration is not constant?

If you meet some problems with non-constant acceleration. Just follow the above steps, take integration to get the equations first, then substitute the values to find out the answers. Here is an example for a = 24t. Try to find the displacement after 3 seconds. The object is originally moving at 4m/s.

Other non-constant acceleration problems can be solved using this method.