In the physics world, velocity (v), position (x), acceleration (a) and time (t) are the four key ingredients in solving equations of motion. You may get the acceleration, initial velocity (v_{0}) and elapsed time of a particle and have to solve for the final velocity (v_{f}). A variety of other permutations applicable to countless real-world scenarios are possible. These concepts appear in four essential equations:

1. x = v_{0}t + (1/2)at^{2}

2. v_{f}^{2} = v_{0}^{2} + 2ax

3. v_{f} = v_{0} + at

4. x = (v_{0}/2 + v_{f}/2)(t)

These equations are useful in calculating the speed (equivalent to velocity for present purposes) of a particle moving with constant acceleration at the moment it strikes an unyielding object, such as the ground or a solid wall. In other words, you can use them to calculate impact speed, or in terms of the above variables, v_{f}.

## Step 1: Assess Your Variables

If your problem involves an object falling from rest under the influence of gravity, then v_{0} = 0 and a = 9.8 m/s^{2} and you need only know the time t or the distance fallen x to proceed (see Step 2). If, on the other hand, you may get the value of the acceleration a for a car traveling horizontally over a given distance x or for a given time t, requiring you to solve an intermediate problem before determining v_{f} (see Step 3).

## Step 2: A Falling Object

If you know an object dropped from a rooftop has been falling for 3.7 seconds, how fast is it going?

From equation 3 above, you know that v_{f} = 0 + (9.8)(3.7) = 36.26 m/s.

If you are not given the time but know that the object has fallen 80 meters (about 260 feet, or 25 stories), you would use equation 2 instead:

v_{f}^{2} = 0 + 2(9.8)(80) = 1,568

v_{f} = √ 1,568 = 39.6 m/s

You're done!

## Step 3: A Speeding Car

Say you know that a car that started from a standstill has been accelerating at 5.0 m/s for 400 meters (about a quarter of a mile) before driving through a large piece of paper set up for a celebratory display. From equation 1 above,

400 = 0 + (1/2)(5)t^{2}

400 = (2.5)t^{2}

160 = t^{2}

t = 12.65 seconds

From here, you can use equation 3 to find v_{f}:

v_{f} = 0 + (5)(12.65)

= 63.25 m/s

## Tip

Always use an equation first for which there is only one unknown, which is not necessarily one that contains the variable of ultimate interest.