Velocity of a particle problem

[andycampbell1]

Homework Statement

Hi just having a problem with this question not sure how to start if off because I only have two variables instead of three. I also think I am working out the velocity wrong. I have solutions for the questions.
The displacement of a particle is given by the expression s = 2t^3 - 3t^2 + 6t - 5 (mm)
Determine (i) The velocity after 2 s. Answer v=18mm/s
(ii) The acceleration after 3s Answer a=30mm/s^2

Homework Equations

v= u+at
s=ut +1/2 at^2
v^2 = u^2+2as

velocity = s / t
acceleration =change in v/ time

The Attempt at a Solution

I have put the time 2s into the equation above and have came out with displacement of 11.
I have tried to use the formula v = s/t, 11/2 = 5.5. Where have I went wrong have I used the wrong equation or is there something I am missing.

Thanks in advance.

 

[da_nang]

Since the displacement isn't uniform, you're looking for the slope of the tangent to that function at that point in time. How would you go about finding it?

 

[gneill]

The usual kinematic equations that you list under Relevant Equations apply when the motion is governed by constant acceleration. In this case you are given a function representing the displacement with respect to time that contains cubic, square, and linear components, making the acceleration vary with time.

This means that you'll have to fall back on the fundamental relationships between position, velocity, and acceleration, namely how they are related through differential calculus; v = ds/dt, a = dv/dt.

 

[andycampbell1]

Sorry I don't really get this. My notes say that if displacement is a function of time, then velocity will be a function of displacement, i.e. v or w = rate of displacement. Hence, acceleration will be a function of velocity, i.e. a = rate of change of velocity.i.e. s=f(t) therefore v=f(s) = ds/dt

also a = f(v) = dv/dt = d^2s/ dt^2.

So say for instance, I am getting 11 as my displacement from the equation at the beginning and v=f(s) = ds/dt v seems to me as if it would be 11/2. I just can't think of any other way that this could be done.

 

[gneill]

You have a function for the displacement with respect to time. Differentiate it with respect to time to find the function for the velocity with respect to time. Differentiate again to find the acceleration with respect to time.