# In the equation for acceleration (with known $v$, $u$ and $s$) why velocity is squared and displacement is multiplied by 2? [closed]

In the following equation $$a = \frac{v^2 − u^2 } {2s} ,$$ where $$v$$ is the final velocity, $$u$$ is the initial velocity, and $$s$$ is displacement. Why is velocity squared and displacement multiplied by 2?

## closed as off-topic by Aaron Stevens, Jon Custer, GiorgioP, stafusa, ZeroTheHeroMar 8 at 17:44

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Aaron Stevens, Jon Custer, GiorgioP, stafusa, ZeroTheHero
If this question can be reworded to fit the rules in the help center, please edit the question.

• Have you tried to derive this formula? Start from $v=u+a t$ and $s=u t+\frac{1}{2}a t^2$. – G. Smith Mar 7 at 18:14
• In these simpler equations for how velocity and displacement depend on time, do you understand why the second term in the second one has $t$ squared and a factor of 1/2? These two things, plus algebra, lead to the velocities being squared in your equation, and the displacement being multiplied by 2. – G. Smith Mar 7 at 18:25
• If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Mar 7 at 18:47

## 1 Answer

The kinematic equation for constant acceleration $$a = \dfrac{(v^2 − u^2 )} {2s}$$ can be changed to the following equation where $$m$$ is the mass of the object and $$F$$ is the force acting on the object.

$$\frac 12 m v^2 - \frac 12 m u^2 = ma\, s = F\, s$$

The left hand side is the change in the kinetic energy of the object and the right hand side is the work done on the object by an external force.

• I don't think this is a worthwhile direction for this kinematics question. G Smith's comment is far more likely the right way to go. – Kyle Kanos Mar 8 at 11:10