The velocity of a particle moving along the x-axis is given by v(t) = √t - cos(e^t) for t is greater than or equal to 0. Which of the following statements describes the motion of the particle at t=1?

Question

A: The particle is moving to the left with positive acceleration

B: The particle is moving to the right with positive acceleration

C: The particle is moving to the left with negative acceleration

D: The particle is moving to the right with negative acceleration.

Answer 1

Hi Joe C.,

Here you are taking a calculus course. So use the concepts of calculus (and physics): velocity is a vector quantity, so you can directly plug into the equation to find out if it is positive (motion to right) or negative (motion to left) at the given time. Acceleration is the rate of change of the velocity, so that's the first derivative of the velocity. Take the derivative of the velocity equation with respect to t, and then evaluate it at the given value for t. Acceleration will be positive (towards the right) or negative (towards the left). Notice that I've used "towards" for acceleration, since you aren't necessarily moving TO the right (velocity) when your acceleration is TOWARDS the right.

You know how to use the chain rule for derivatives, right? For example, d(-cos(e^t))/dt [that's a "nested" function, so progressively unnest it!] = -(d(cos(e^t))/dt [1st level of unnesting] = -(-sin(e^t)d(e^t)/dt [2nd level of unnesting] = sin(e^t)(e^t) [3rd level of unnesting]

Hope that's sufficient to guide you,

-- Cheers, -- Mr. d.