A particle moves along a straight line. Its displacement at time 𝑡 is 𝑥 equals sin 𝑡. Which of the following statements about the acceleration of the particle is true? Is it (A) it’s equal to negative 𝑥, (B) it is equal to the velocity of the particle, (C) it’s equal to negative 𝑣, where 𝑣 is the velocity of the particle, or (D) it’s equal to 𝑥?
We’re given information about the displacement of a particle at time 𝑡. And we’re looking to form an expression for the acceleration. So let’s recall how we can link these two. Firstly, we know that velocity 𝑣 is change in displacement with respect to time. In other words, given an expression for 𝑥, the displacement of the object, we can differentiate it with respect to 𝑡 to find an expression for 𝑣.
Similarly, acceleration is rate of change of velocity. It’s therefore d𝑣 by d𝑡. Since velocity itself is the derivative of displacement with respect to time, we can say that acceleration must be the second derivative. So we’ll begin by differentiating our expression for displacement once to find the velocity and then a second time to find the acceleration.
Now, to do this, we have 𝑥 equals sin 𝑡. And so we can recall a cycle that helps us to differentiate sine and cosine functions. The first derivative of sin 𝑥 with respect to 𝑥 is cos 𝑥. Then when we differentiate cos 𝑥, we get negative sin 𝑥. The first derivative of negative sin 𝑥 is negative cos 𝑥. And differentiating negative cos 𝑥 gives us sin 𝑥.
Since velocity is the first derivative of 𝑥 with respect to 𝑡, we need to differentiate sin 𝑡. And so we find that 𝑣 is equal to cos 𝑡. Then we differentiate this expression with respect to 𝑡 to find the expression for acceleration. The derivative of cos 𝑡 is negative sin 𝑡. So acceleration is negative sin 𝑡.
And we now have three expressions that describe the motion of the particle. We have velocity, displacement, and acceleration. Now, in fact, if we compare acceleration and displacement, we see that one is the negative version of the other. 𝑎 is equal to negative 𝑥 and vice versa. The correct answer is therefore (A) the acceleration is equal to negative 𝑥.