Q:

Can an object *immediately* start moving at a high velocity?

What I mean is, suppose a ball is fired from a cannon. Suppose the ball is moving at 100 m/s in the first second. Would the ball have started from 1m/s to 2m/s and gradually arrived at 100m/s? And is the change so fast that we are not able to conceive it? Or does the ball actually start its motion at 100m/s as soon as the cannon is fired? 
Suppose a 40-wheeler is moving at the speed of 100 kmph. And it collides with a car and does not brake (The car is empty in my hypothesis ;-) ). If a body does not reach a certain speed immediately and only increases speed gradually, does that mean that as soon as the truck collides with the car, the truck momentarily comes to rest? Why I ask this is because the car is not allowed to immediately start at the truck's speed. So when both of them collide, the car must start from 0 to 1kmph to 2kmph and finally reach the truck's speed. Does this not mean that the truck must 'restart' as well? Intuition tells me I'm wrong but I do not know how to explain it physically being an amateur.
When is it possible for a body to accelerate immediately? Don't photons travel at the speed of light from the second they exist?

A:

The answer to your question is "No" as this would require an infinite acceleration and hence an infinite force to be applied.
However to simplify problems it is sometimes convenient to assume an instantaneous jump in velocity as in the left hand graph.  

It might be that the actual change in velocity is as per the green line in the right hand graph but because the change in velocity $\Delta t$ takes place over such a short period of time, perhaps $\Delta t \ll 1$ in this example, the approximation to an "instantaneous" change has little bearing on the final outcome.
So every time you see a velocity against tine graph with some sort of "corner", where a gradient (= acceleration) cannot be found, you have to think that the corner is actually rounded but that rounding occurs over a very short period of time compared with the time scale of the whole motion it matters not a lot.
Another example is of problems where a ball rebounds from the ground and you have to find the time it takes to reach a certain height after the rebound.
It is unlikely that you consider the time that the ball is in contact with the ground and slowing down with an acceleration much greater than $g$ (acceleration of free fall) stopping and then initially accelerating upwards at an acceleration greater than $g$.
Usually the sums are done assuming that the acceleration is $g$ all the time because the time that the ball is in contact with the ground is so much smaller than its time of flight through the air.

A:

To answer this, one must understand how motion starts. The motion of the canon-ball starts due to force applied on it. The force directly does not lead to a high velocity. Force leads to acceleration, ie, change in velocity. So, the motion is dictated by
$$F=ma=m\dfrac{\mathrm dv}{\mathrm dt}$$
The change in velocity $\mathrm dv$ during the first moments (which is the question) depends on the infinitesimal time period $\mathrm dt$ you are considering. This time period can be arbitrarily small, and as the force $F$ is fixed, the change in velocity $\mathrm dv$ can be arbitrarily small too. So yes, the motion starts 'gradually', which is to say that the ball doesn't have $0$ velocity and then $100\ \mathrm{m/s}$ without reaching a velocity in between.
Note: We should technically be talking about changes in momentum, instead of velocity. If one adds quantum and relativity to the picture, the analysis becomes more complicated, and the question itself has no meaning in that scenario. I assume you only wanted to know about everyday 'canon-ball' situations.

A:

Nothing propagates instantly
For starters, any real object has a non-zero size. If you apply force to one side of a cannonball, it will start moving before the other side of the cannonball can be affected by that force. The time will be very small ("so fast that we are not able to conceive it") but nonzero. The cannonball will experience some compression, and in general (for reasonable energies) the propagation will be limited by the speed of sound in that material.
So even without even going to gradual acceleration (the main purpose of the cannon barrel - there's a reason why they're so long!), even a simple transfer of momentum by a single impact (e.g one Newtonian ball hitting another) isn't instant.

