Thread: Particle problem

The position of a particle moving along an x axis is given by x = 14.0t2 - 6.00t3, where x is in meters and t is in seconds. Determine (a) the position, (b) the velocity, and (c) the acceleration of the particle at t = 7.00 s. (d) What is the maximum positive coordinate reached by the particle and (e) at what time is it reached? (f) What is the maximum positive velocity reached by the particle and (g) at what time is it reached? (h) What is the acceleration of the particle at the instant the particle is not moving (other than at t = 0)? (i) Determine the average velocity of the particle between t = 0 and t = 7.00 s.

I was able to get letter a, but i am not sure what i need to do to get the rest, do i need to take the derivative or simply use regular velocity formulas? thank you

(b) The velocity is the derivative of position with respect to time

(c) The acceleration is the second derivative of position with respect to time

(d) To find local extrema of functions we try to find the stationary points (points where the derivative = 0) and then confirm that they are maxima asserting that the second derivative < 0. Since this function happens to have a single local extreme it is an easy task

(e) This is equivalent to finding the 't' that maximizes the function. You find this while solving (d)

(f) The same as before but now dealing with velocity instead of position.

(g) Same as e

(h) The instant the particle isn't moving => v=0

(i) I hope you know (intuitively) how to find the average velocity of something

You can't use the "regular velocity formulas" as these suppose a constant velocity which this is plainly not.

Since this is a calculus forum, I presume you are expected to use calculus.

So differentiate what you have w.r.t t and that gives you the velocity.

Differentiat that w.r.t t to get the acceleration.

Plug in t=7 to get the acceleration at that time.

To get the max pos coord work out where the velocity is zero and check that at that point the acceleration is negative.

Solve the equation in t for the time when that happens.

Work out the time at which the velocity is zero by solving a quadratic which you can get from where you derived the velocity. Then feed that time into the formula for acceleration.