# Calculating time using SUVAT equations with unknown initial velocity

I understand that the equation used to find the equation used to find displacement when the initial velocity is unknown is:

$vt-(at^2)/2$

how would you rearrange this to solve for $t\;?$ When I try I get $t$ on both sides of the equation.

• Quadratic formula maybe? Jan 24, 2016 at 12:52
• Shouldn't it be "when the initial velocity is known"? Jan 25, 2016 at 14:49

You can rearrange using the quadratic formula to get: $$t = \frac{-u±\sqrt(u^2+2as)}{a}$$ Thanks AccidentalFourierTransform for the answer.

The equation used to find the displacement in motion with constant acceleration in 1 dimension is -

$s = ut + \frac{a{t}^{2}}{2}$

where u is the initial velocity , s is displacement and a is acceleration (constant).

You can re-write the equation as -

$t^2 + \frac{2u}{a}t - \frac{2s}{a} = 0$

or $(t + \frac{u}{a})^2 - \frac{2s}{a} - \frac{u^2}{a^2} = 0$

$(t + \frac{u}{a})^2 = \frac{u^2 + 2sa}{a^2}$

$t + \frac{u}{a} = \pm\frac{\sqrt{u^2 + 2sa}}{|a|}$

So, $t = - \frac{u}{a} \pm\frac{\sqrt{u^2 + 2sa}}{|a|}$

(You can get the same result by using the quadratic formula.)

• The OP specified that $u$, the initial velocity is unknown... Jan 24, 2016 at 16:17