Why is a particle at rest while accelerating?

[bourne]

Homework Statement

consider a particle moving in a straight line and assume that its position is defined by the equation,
x= 6(t^2) - (t^3)

Homework Equations

v=dx/dt= 12t - (3t^2) and
a=dv/dt= 12-6t

The Attempt at a Solution

the question is at t=0, x=0 at the origin the velocity is zero while the acceleration is 12 units. How's it possible that a particle is having zero velocity while it is accelerating? i agree mathematically, but wat about the physics?
suppose if i consider instantaneous velocity.
it's limit t tends to zero dx/dt.
dx=infinitesimally small distance= x(at position 2) - x(position 1)= x(2) - x(1)
x(1)=0, that is origin. if x(2) is very very close to x(1), then the limit t tends to zero dx/dt will become zero that is instantaneous velocity will become zero.
But in that very very small time 'dt', if particle did travel infinitesimally very small distance 'dx', such that the instantaneous velocity is zero, then how could there be acceleration in that same interval of time 'dt' ?

 

[Gokul43201]

A ball is throw vertically up. What is its velocity at the top? What is its acceleration?

 

[learningphysics]

Even though dx = 0 over some time dt... doesn't mean dv = 0 over that same time dt...

 

[bourne]

Even though dx = 0 over some time dt... doesn't mean dv = 0 over that same time dt...

if a particle didn't move the distance dx, how can it have velocity then?
m pretty confused. Is there any mathematical justification?
is thr any mechanics stuff that u want me to go through so that i can make myself clear?

 

[Gokul43201]

if a particle didn't move the distance dx, how can it have velocity then?

It can not, but it can have an acceleration. dv is related to acceleration, not velocity.

m pretty confused. Is there any mathematical justification?

Yes. There is no reason that a function have zero slope at f(x)=0.

 

[bourne]

A ball is throw vertically up. What is its velocity at the top? What is its acceleration?

this example is when we analyse things in the gravitational field.
Its a kind of conservative field. if a particle is placed anywhere near the surface of the Earth it possesses potential energy by virtue of its position.

Similar is the electromagnetic field.

But suppose we consider a fluid flow field like fluid flow in a pipeline or boundary layer flow over the aircraft, where the force due to gravity is canceled by the boyant force of the fluid particle, fluid flows across two points due to the pressure difference, i come across the functions of position , velocity and accelerations, wherein after solving i find that particle velocity is zero while it has acceleration.

I dun't know whether u guys r geting me or not, anyways thanks for being patient and helping me out.

 

[learningphysics]

Let's assume that 0 velocity => 0 acceleration. Suppose an object is at rest. Is motion possible for this object? Assume motion happens at some time t = 0. so for t<0 we have 0 velocity. But if motion happens at t = 0, then v>0 at t = 0... but that means there was an acceleration at some time t<0. but at t<0 we have 0 velocity and hence 0 acceleration according to our initial assumption...

contradiction. so no motion from rest is impossible.

0 velocity => 0 acceleration leads to the conclusion that motion from rest is impossible.

 

[bourne]

okay i get ya