A particle is moving in a straight line such that its displacement 𝑥 after 𝑡 seconds is given by 𝑥 equals two 𝑡 squared minus 24𝑡 minus 15 meters for 𝑡 is greater than or equal to zero. Determine the time after which the particle changes its direction.
We’ve been given an expression for displacement, 𝑥, at 𝑡 seconds. So, we need to ask ourselves how we’re going to link that equation to the direction in which the particle is traveling. Well, we need to consider the velocity of the particle. Velocity is a vector quantity. It can be positive or negative, depending on the direction in which the particle is traveling. We also know that given an expression for displacement, 𝑥, we can differentiate 𝑥 with respect to 𝑡 to find an expression for velocity.
So, let’s begin by differentiating our expression for 𝑥. It’s made up of three terms. So, we’ll differentiate term by term. We recall that to differentiate a power term, we multiply the entire term by the exponent and reduce that exponent by one. So, when we differentiate two 𝑡 squared, we get two times two 𝑡. Then, the derivative of negative 24𝑡 is negative 24. We also know that the derivative of a constant is zero. So, 𝑣 is two times two 𝑡 minus 24, and the velocity is four 𝑡 minus 24.
We said velocity can be positive or negative, depending on the direction in which it’s traveling. And so, the particle will change direction when its velocity changes from positive to negative or negative to positive, when it changes sign. Well, it follows that, at this point, its velocity must be equal to zero. So, we can say zero is four 𝑡 minus 24. We solve for 𝑡 to find the time at which the particle changes direction. So, we add 24 to both sides, such that four 𝑡 is 24. Finally, we divide through by four. 𝑡 is equal to 24 divided by four, which is equal to six or six seconds.
We can, therefore, say the time after which the particle changes its direction is 𝑡 equals six seconds.