There are several other formulas that are very useful when the acceleration is uniform.

• Do not use these equations if the acceleration is changing!
• These other formulas are based on combinations of the basic velocity and acceleration formulas.

Formula 1

A common problem is to figure out the final velocity of an object after it has been accelerating for a certain time.

This is not a "new" formula, but just a manipulation of the standard acceleration formula.

v_f = v_i + at

Example1: I’m driving my Camaro at 61km/h when I notice that there is a school zone ahead. If I slam on the brakes for 6.7s and experience acceleration of 1.5m/s², determine if I will be under the 30km/h posted speed limit.

First, change the initial velocity into m/s…

v_i = 61km/h = 16.9444444m/s

Remember to keep all these extra digits on your calculator, but keep track of the actual sig digs.

Keep in mind that since I was slowing down, the acceleration is negative.

Now use the formula. This one is so common, you don’t even need to show how you manipulated it if you remembered it.

v_f = v_i + at

= 17m/s + (-1.5m/s²)(6.7s)

= 6.894m/s = 24.81 km/h

v_f = 25 km/h

Yep, I slowed down enough. Notice that I kept track of sig digs and changed it over to km/h at the end so I coould compare it to the posted speedlimit easily.

Formula 2

We will also do problems where we need to calculate the displacement of an accelerating object after a time interval has passed and we know its initial velocity.

d = v_it + ½ at²

• Be careful with this formula. Only the time is squared in the last term, not acceleration and time.
• As a bonus, a lot of the time v_i will be zero, which cancels out the first term and leaves you with…

d = ½ at²

Example 2: Occasionally the US Air Force calls me in to test fly their “birds”. A few weeks back I was flying along in my F-22 at 97m/s when I decide to kick in the afterburners for 12.3s. If the afterburners can generate enough thrust to accelerate the F-22 at 26m/s², determine how far I traveled during that time.

d = v_it + ½ at²

= (97m/s)(12.3s) + ½ (26m/s²)(12.3s)²

d = 3.2e3 m

Example 3: I am in a F-22 that is on the runway. From rest, I accelerate the plane at 3.9m/s² for 9.5s to reach take off velocity. Determine how long the runway have to be.

This is an example of a question where the initial velocity is zero (since I’m starting from rest), so…

d = v_it + ½ at²

d = ½ at²

= ½ (3.9m/s²)(9.5s)2

= 1.8e2 m

Formula 3

There is a formula related to formula 2 that can be used when we know the final velocity instead of the initial.

d = v_ft - ½ at²

• Notice that the differences are final instead of initial velocity, and the minus sign instead of addition.
• Otherwise, this formula is used the same way as formula 2.

Formula 4

Another very useful formula is the following…

v_f² = v_i² + 2ad

• Very handy when you have a question with both velocities, acceleration, and displacement.
• Don’t forget to do the square root at the very end if you are solving for a velocity, as the following example shows…

Example 4: Determine the final velocity of a car that starts at 42 m/s and accelerates at 3.78m/s² for a distance of 12 m.

v_f² = v_i² + 2ad

= (42m/s)² + 2 (3.78m/s²) (12m)

v_f² = 1855
Many people leave the answer like this, forgetting that this is velocity squared!

v_f = 43 m/s

Formula 5

The last formula I’ll show you is one of the least used ones.

d = ½ (v_f+ v_i) t

• Only use it when you know for certain that the object has been going through a constant acceleration, even though "a" doesn't appear in the formula.

Example 5: Determine how far a vehicle moved if it started at 12 m/s and accelerated up to 47 m/s in a time of 34s.

d = ½ (v_f+ v_i) t

= ½ (47 m/s + 12 m/s) (34s)
Notice that it really doesn’t matter which order the velocities are added in.

d = 1.0e3 m

How to Choose the Right Formula!

So, how do you figure out which formula to use for a particular problem?

• As you look back through the formulas, you'll see that of the five basic things we measure about the motion of an object (v_f , v_i , a , t , and d) , each formula only has four.
• To choose the correct formula, look for the one thing that you are not given or asked for as the "missing" part of a formula.
• The following table may help.

For example, if I had a question where I am given acceleration, displacement, and time, and asked to find initial velocity, I would choose the second formula.

• It’s the only formula that doesn’t have final velocity, which I haven’t been given or asked for.

Remember that for all of these formulas, you may be required to manipulate the formula to find the answer you are looking for.

• Always follow the rule of finding the formula that has all the knowns and unknown that you have, then manipulate it for your unknown, and solve.

Example 6: Determine the displacement of a car that starts at 10 m/s and accelerates at 1.89m/s² and reaches a final speed of 32m/s.

v_f² = v_i² + 2ad
Can you see why I choose this formula?

d = (v_f² - v_i²) / 2a

= (32²-10²) / 2(1.89)

d = 2.4e2 m