A uniformly accelerating particle has a velocity of 2.0 meters per second at a time 𝑡 equals 2.0 seconds and a velocity of negative 7.6 meters per second at a time 𝑡 equals 5.2 seconds. What is the acceleration of the particle?
In this problem, we’re looking to solve for the acceleration of the particle, and we can create a shorthand a that represents the acceleration of the particle we’re trying to solve for. You can see we’re given lots of information about the velocity of the particle at different points in time. And as a first step, we can organize that information in a table so it’s easier to look at and keep track of.
So let’s make a table where we’ll record time in seconds and then the corresponding velocity in meters per second. Now we’re told that at time equals 2.0 seconds, the velocity is 2.0 meters per second. And we’re also told that when the time is 5.2 seconds, our velocity is negative 7.6 meters per second. And again, we want to use this information to solve for the acceleration of the particle. Now we move that information up to the right side of our screen. And again, our mission is to solve for the acceleration that this particle undergoes. So to get there, let’s recall a definition.
You’ll remember that acceleration is defined as a change in velocity and we’ll use the Greek letter delta to represent change. The change in velocity divided by a change in time. And another way of writing that is to say that this would be the final velocity of our particle, 𝑣 sub final minus 𝑣 sub initial, the velocity it starts out with. And all of that is divided by the final time, 𝑡 sub final minus 𝑡 sub initial, the first time we record the velocity for. So now that we have this equation, we can apply it to the data we’re given in our table. So let’s write this out. The acceleration of our particle will be equal to the final velocity of our particle, we have that as negative 7.6 meters per second, minus the initial velocity of our particle, 2.0 meters per second. All of that is divided by 5.2 seconds minus 2.0 seconds. That’s our final time minus our initial time.
Before we calculate these numbers, just a quick thing to notice here. You’ll see that our units on top are meters per second, and our units on bottom in the denominator are seconds. That tells us that our final answer will have units of meters per second per second or meters per second. Now that checks out, that’s the correct unit for acceleration. So we have confidence that we’re on the right track here.
When we start computing these numbers, you’ll see that our numerator is negative 9.6 and our denominator will be 5.2 minus 2.0 or 3.2. And when we punch that into our calculator, we get a final answer of negative 3.0 meters per second squared. That is the acceleration that this particle has undergone between times 𝑡 equals 5.2 seconds and time 𝑡 equals 2.0 seconds.