Question

For the equation $v^2_f = v^2_i +2ad$, how do you solve for $v_i$?

Answer 1

If you look closely, you can see that the above expression is a kinematic equation of motion. From left to right, the expression reads final velocity squared is equal to the sum of initial velocity squared and the product of 2 times acceleration times displacement of an object. Try to isolate $v_i$ from the rest of the terms and arithmetically determine the resulting expression arising from solving it.

Complete Solution:
We are given the kinematic equation of motion $v^2_f = v^2_i +2ad$, where
$v_f$ is the final velocity
$v_i$ is the initial velocity
a is the acceleration of an object, and
d is the displacement of the object.

We are required to obtain an expression that is able to describe the quantitative value of the initial speed $v_i$ of an object.

To do so, we first bring all the terms not involving $v_i$ to one side:
$v^2_i = v^2_f -2ad$

We now solve for $v_i$ by getting rid of its square. We do so by imposing a square root on the entire equation.
$\sqrt{v^2_i} = \sqrt{ v^2_f-2ad }$

$\Rightarrow v_i = \sqrt{ v^2_f-2ad }$ which is the required solution, since $\sqrt{v^2_i} = \pm v_i$, and as $v_i$, by definition, is concerned with only the speed of the object and not its velocity, we consider only the absolute value of $|v_i|$ which denotes its magnitude.

Note:
In addition to the above discussed equation of motion, it is advisable to keep in mind a couple more equations that describe a variety of quantities associated with the motion of objects:
$v_f = v_i +at$, where t is the time taken by the object.
$d=v_i t +\dfrac{1}{2}at^2$
The above equations can be used to describe the motion of an object under various circumstances, correlating the quantities in the equations, and are most often used to determine unknown quantities from a limited description of any object’s trajectory/motion.