### Video Transcript

A particle moves along a straight
line. Its displacement at time ๐ก is ๐ฅ
equals negative cos of ๐ก. Which of the following statements
about the acceleration of the particle is true? Is it (A) it is equal to ๐ฅ? (B) It is equal to negative ๐ฃ,
where ๐ฃ is the velocity of the particle. (C) It is equal to the velocity of
the particle. Or is it (D) it is equal to
negative ๐ฅ?

In this question, weโve been given
information about the displacement of a particle time ๐ก. And weโre looking to find
information about the acceleration of that same particle. And so we recall the link between
acceleration and displacement. Acceleration is the rate of change
of velocity of the particle. And the velocity itself is the rate
of change of displacement. So, we differentiate an expression
for displacement with respect to time to get us an expression for velocity. And then we differentiate once more
to get an expression for acceleration.

So, to find an expression for the
acceleration of the particle, weโre going to differentiate negative cos of ๐ก with
respect to ๐ก. In fact, thereโs a cycle that can
help us remember how to differentiate trigonometric functions. The derivative of sin ๐ฅ is cos
๐ฅ. Then, the derivative of cos ๐ฅ is
negative sin ๐ฅ. If we differentiate negative sin ๐ฅ
with respect to ๐ฅ, we get negative cos ๐ฅ. Then, if we differentiate negative
cos ๐ฅ with respect to ๐ฅ, we get back to sin ๐ฅ. So, letโs begin by differentiating
our expression for ๐ฅ with respect to time ๐ก. This tells us that the velocity is
the derivative of negative cos ๐ก. We can see from our cycle that the
derivative of negative cos ๐ฅ is sin ๐ฅ. So, the derivative of negative cos
๐ก with respect to ๐ก is sin ๐ก.

To find an expression for
acceleration, weโre going to differentiate our expression for velocity with respect
to time. Once again, we see from our cycle
that the derivative of sin ๐ฅ with respect to ๐ฅ is cos ๐ฅ. And so this means that the
derivative of sin ๐ก with respect to ๐ก is cos ๐ก. So, we have three expressions
describing the motion of the particle. Velocity is sin ๐ก, acceleration is
cos ๐ก, and ๐ฅ, displacement, is negative cos ๐ก.

We can see that, in fact, our
expressions for acceleration and displacement look quite similar. However, they are negatives of one
another. So, we can say that ๐ฅ is the
negative of the acceleration or vice versa. ๐ is equal to negative ๐ฅ. Going back to the options given to
us in this question, we see that that is equivalent to (D). The answer is (D). It is equal to negative ๐ฅ.