I've encountered a strange kinematics problem in a textbook.

Rest assured, I have long finished my studies in Physics, I simply decided to get back to the basics in order to refresh my skills.

The problem is the following:
"A particle is moving along the X-axis. The coordinate of the particle is changing with relation to time according to the law: $x = c_1t + c_2t^3$, where $c_1 = 2 m/s$, $c_2 = 1 m/s^3$.
a) The velocity and acceleration of the particle at a point of time = 2s
b) The mean velocity with which the particle travels in the first 3s"

The way I see it I can sort of unofficially take $c_2$ as the jerk, $c_1$ as the initial velocity and, maybe, the initial acceleration is = 0, even though nothing is mentioned about that. So the acceleration at 2s should be $2m/s^2$ and velocity should be $3m/s$ but the answer in the textbook is $v = 14m/s$ and $a = 12m/s^2$. I don't even want to talk about b).

I have no idea how this is supposed to be solved, one because I don't see any relevant information in the problem description and, two, because nothing like this was covered in the lesson of this problem.


closed as off-topic by John Rennie, Bill N, stafusa, Kyle Kanos, Emilio Pisanty Oct 2 '17 at 18:39

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  • 3
    $\begingroup$ Calculus! mathgenius? $\endgroup$ – Bill N Oct 1 '17 at 16:44
  • 2
    $\begingroup$ @BillN, the name is supposed to be ironic. $\endgroup$ – mathgenius Oct 1 '17 at 18:13
  • $\begingroup$ Regardless of whether this is formal homework assigned to you by an instructor, our guidelines for homework and exercise questions apply. Please read them carefully. $\endgroup$ – Emilio Pisanty Oct 2 '17 at 18:39

This is actually very simple. On part (a), the question actually asked for the instantaneous velocity and acceleration of the particle at time 2 sec. So what you have to do is to differentiate the function of the position with respect to time once to get the velocity and once more to get the acceleration function. From there you just have to plug in the value of t=2s and you will get the same answer from the textbook.

For part (b), you have to remember that mean/average velocity is the total displacement divided by the total time. So you just have to plug in t=3s into the displacement function. You will get x(3) = 2(3)+(1)(3)^3 = 33m. Mean velocity = 33m/3s = 11m/s.


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