An object moving along a line has a velocity 𝑣 of 𝑡 equals five 𝑡 sin of 𝑡 plus two times the natural log of 𝑡 plus five, where 𝑡 is greater than or equal to zero and less than or equal to eight. How many times does the object reverse direction?
Here, we’ve been given a function for velocity in terms of time. So let’s think about what we know about our function for velocity. We know that velocity is a vector quantity. It has both a direction and a magnitude, unlike speed which just has a magnitude. If the particle was moving along a horizontal line, we could say that if the velocity was greater than zero, then it’s moving to the right. And if the velocity is less than zero, then the object is moving to the left. In this case then, a change in sign indicates a change in direction.
Now, we don’t know whether our object is moving on a horizontal line. But we do know it’s moving along a line. So we can say that when the velocity changes sign, the direction of the object is reversed. So how do we establish when the velocity changes sign? The easiest way is simply to use our graphical calculators to sketch the graph of 𝑦 equals five 𝑡 sin of 𝑡 plus two times the natural log of 𝑡 plus five. If we plot this in the closed interval zero to eight, we see that the graph of 𝑣 of 𝑡 equals five 𝑡 sin of 𝑡 plus two times the natural log of 𝑡 plus five looks a little something like this.
We’re looking to find the number of times that the sign of the velocity changes. In other words, it goes from positive to negative or vice versa. At a little over 𝑡 equals three, the velocity does indeed change from positive to negative. And at a little over six, it changes again from negative to positive. And we can say that the velocity changes from positive to negative and negative to positive twice in the closed interval zero to eight. And therefore, the object reverses direction twice in this time.