Understanding Kinematics: Uniformly Accelerated Motion Equation Explained

[LuGoBi]

Well, I think this one is pretty simple, but still, I don't know how to solve it.

We all know that for uniform velocity in a straight line the following equation sets the relationship between time and distance traveled: S = So + Vt (Eq. 1)

When it comes to uniformly accelerated motion, the only difference is that the velocity is changing constantly, according to the following equation: V = Vo + at (Eq. 2)

Now, if you insert Eq. 2 in Eq. 1 you get: S = So + Vot + at^2 (Eq. 3)

But we all know the correct equation is S = So + Vot + at^2/2! Besides, the second derivative of Eq. 3 is 2a, when the correct one is, by definition, a, obviously. So it's clearly wrong. What's the deal with this?

 

[Doc Al]

In Eq. 2, V stands for the final speed after some time. To use it in Eq. 1, you'd have to replace V by the average speed, since that equation only applies for constant speed (or average speed). Since the acceleration is uniform, the average speed is just (Vo + Vf)/2 = (Vo + Vo + at)/2 = Vo + at/2. Plug that into Eq. 1 and see what happens.

 

[LuGoBi]

Damn it, that's beautiful. Thank you very much.

 

[mr.survive]

The reason for the wrong answer is that ;
v = u + at is equation to find final velocity in constant acceleration, and you are putting this final velocity in 2nd equation (BUT VELOCITY IS CHANGING AT EVERY POINT)

SO,YOUR EQUATION MODIFIES AS:-

S = So + v1t1 + v2t2 + ...(where t1 + t2 +...= t)

so,in general,u MUST USE,

dS = v.dt