Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Video: Analyzing the Kinematics of an Object with a Position Given by a Quadratic Function of Time

Ed Burdette

A particle moves along the 𝑥-axis according to 𝑥(𝑡) = 10𝑡 − 2𝑡² m. What is the instantaneous velocity at 𝑡 = 2 s? What is the instantaneous velocity at 𝑡 = 3 s? What is the instantaneous speed at 𝑡 = 2 s? What is the instantaneous speed at 𝑡 = 3 s? What is the average velocity between 𝑡 = 2 s and 𝑡 = 3 s?

09:21

Video Transcript

A particle moves along the 𝑥-axis according to 𝑥 as the function of 𝑡 equals 10 times 𝑡 minus two times 𝑡 squared meters. What is the instantaneous velocity at 𝑡 equals two seconds? What is the instantaneous velocity at 𝑡 equals three seconds? What is the instantaneous speed at 𝑡 equals two seconds? What is the instantaneous speed at 𝑡 equals three seconds? And what is the average velocity between 𝑡 equals two seconds and 𝑡 equals three seconds?

The first thing we can notice here about all these different questions is that we’re talking about velocity in the first two, speed in the second two, and then an average velocity in the third. And we’re given different values of time at which we wanna solve for these terms. Let’s start by writing a little bit of shorthand for each of these five questions. We can call the instantaneous velocity at 𝑡 equals two, we’ll write that as 𝑣 of two. And likewise, we can write the instantaneous velocity at 𝑡 equals three seconds as 𝑣 of three. Now for the instantaneous speed at 𝑡 equals two seconds, let’s call that 𝑠 of two. And our instantaneous speed at 𝑡 equals three seconds, let’s call that 𝑠 of three. Finally, let’s give a name to our average velocity. Let’s call that 𝑣 sub avg, for average.

Alright, here we go. We’ve got our defining equation for motion right here. And over on our right-hand side of the screen, we see all the five values we wanna solve for. Now the first thing you may notice is that we’re asked for velocity as a function of time, but our function is position as a function of time. And we can tell that based on the units written here, units of meters. So our first step is to convert this equation to one that tells us what the velocity of this object is, as a function of time. To do that, we can recall a definition for velocity. Velocity as a function of time is defined as the change in a particle’s position, we’ll use the Greek symbol Δ for change, so the change in the particle’s position as a function of time divided by the change in time elapsed.

Mathematically, what this means is if we want velocity as a function of time, we can take our position as a function of time equation and take the derivative of that equation with respect to time. That means we will take the derivative with respect to time of the equation we’ve been given for position as a function of time. In other words, velocity is a function of time is equal to the time derivative of 10𝑡 minus two 𝑡 squared. Taking the derivative of that function with respect to time, we end up with an equation for velocity as a function of time of 10 minus four 𝑡 meters per second. This is the equation we’ll use to solve for the five different values that our question asks for.

Let’s start off by solving for velocity when time equals two seconds. So if we plug in two for 𝑡 everywhere we see it in this equation, we find- we get 10 minus four times two meters per second. That’s 10 minus eight which equals two meters per second. That is our instantaneous velocity when time equals two seconds. We’ve now solved for 𝑣 when 𝑡 equals two seconds, so we can cross that off our list.

Now we move on to 𝑣 when 𝑡 equals three seconds. Plugging in three every time 𝑡 appears in this equation, we find 10 minus four times three meters per second. This equals 10 minus 12 which is negative two meters per second. That’s our instantaneous velocity when time equals three seconds.

Now we get to an interesting point because we’re no longer solving for instantaneous velocity, but we’re solving for instantaneous speed. Now remember that speed is a scalar quantity. So when it comes to our equation, speed as a function of time will equal velocity as a function of time within absolute value bars. Speed is the magnitude of velocity working off of this equation for velocity as a function of time. What this means is that when we wanna solve for speed when 𝑡 equals two seconds, all we need to do is put absolute value bars around the same equation for velocity when 𝑡 equals two seconds. We can see that when 𝑡 equals two seconds, 10 minus eight is already a positive number, so we need not worry about the absolute value bars. And speed, when time equals two seconds, is equal to two meters per second. So our instantaneous speed at two seconds is equal to our instantaneous velocity at two seconds.

Now let’s move on to the instantaneous speed when time equals three seconds. So we again use our absolute value bars, 10 minus four times three, close up with our absolute value bars, and complete with our units, meters per second. Now as you look into that equation, you see that we’ve got 10 minus 12 or negative two. But because of our absolute value bars, that flips the negative to a positive. And we wind up with an answer of positive two meters per second. So in this case, our instantaneous speed at three seconds is not equal to our instantaneous velocity at three seconds. Thanks again to our absolute value bars.

Now we’re left with just one more thing to solve for, 𝑣 sub avg, the average velocity of the particle between time equals two seconds and time equals three seconds. We can start by defining what average velocity is. Mathematically, the average velocity between two points in time is equal to the velocity at that initial point in time plus the velocity at the final point in time divided by the change in time. When we write this out for our own given values, we see we’ve already solved for 𝑣 of two, that’s two meters per second. And we’ve also already solved for 𝑣 of three, that’s negative two meters per second. Now the time involved is between 𝑡 equals two seconds and three seconds. So our change in time is one second. Now take a look at the numerator in this equation. We have positive two meters per second plus minus two meters per second. So what we get is: zero meters per second is our average velocity between times 𝑡 equals two seconds and 𝑡 equals three seconds. This concludes all the values we wanted to solve for.