Video Transcript
A particle moves in a straight line such that at time 𝑡 seconds, its velocity is given by 𝑣 equals two 𝑡 squared minus 𝑡 minus six meters per second for 𝑡 is greater than or equal to zero. After how many seconds does the direction of the particle’s motion change?
Let’s begin by recalling what we understand about the word “velocity.” Velocity is the rate of change of the position or displacement of an object. It’s a vector quantity such that it has both magnitude and direction. If we think purely about the magnitude of velocity, we know that that’s equal to speed. The keyword in our definition here is this word “direction.” Since velocity has a direction, we can say that it can be both positive or negative. And we can therefore say that an object will change direction when its velocity changes sign, in other words, when it changes from negative to positive or vice versa. We can find the instantaneous moment when this occurs by finding the exact moment that the velocity is actually equal to zero.
Our expression for velocity is two 𝑡 squared minus 𝑡 minus six. So let’s set that equal to zero and solve for 𝑡. This is a quadratic equation. So we’re going to begin by factoring the expression on the left-hand side. We know it will factor into two pairs of parentheses. We also know the first terms must be two 𝑡 and 𝑡 since their product gives us two 𝑡 squared. We need the other two terms to multiply to make negative six. And when we distribute our parentheses, we need to have negative one 𝑡. And so we get two 𝑡 plus three times 𝑡 minus two equals zero. Now, of course, for the product of these two expressions to be equal to zero, either two 𝑡 plus three must itself be equal to zero or 𝑡 minus two must be equal to zero.
So we’ll solve these equations for 𝑡. We begin to solve this first equation by subtracting three from both sides, then dividing through by two, giving us 𝑡 equals negative three over two. Now, we were told that 𝑡 must be greater than or equal to zero. So we’re going to disregard this value of 𝑡. And so we solve our second equation for 𝑡. This time, we add two to both sides, and we get 𝑡 equals two. Since this is the positive value of 𝑡 that makes our velocity equal to zero, we know that this is the time at which the particle’s motion changes direction.
The answer is 𝑡 equals two seconds.
