# Is Acceleration/Decceleration smooth?

• Stephen T
In summary: There are a few ways to do this. One is to use approximations. Another is to use a different kind of mathematics that doesn't involve discontinuities. A third is to use models of the real world.
Stephen T
Firstly I must say if this is in a wrong sub-forum I apologise. This is my first post and I'm new to the website so please bare with me. Also I was unsure of the Prefix so I again apologise if that to, was incorrect.

But my Question is that is the acceleration (or deceleration) of an object smooth?
By this I mean can the speed of an object increase instantaneously (for example) from 2Mph to 4Mph?
Or must it be smooth in it's acceleration by passing all speeds in-between 2Mph and 4Mph to get to the end 4Mph?

By passing all speeds in-between i mean such as it goes (in terms of speed) through every number in-between the 2 and 4Mph.

Sorry if this is not worded well as I am unsure of how to, but I hope you understand and reply.

In principle, acceleration can be instantaneous - hit it with a hammer. In practice, it always happens over some time - e.g. the steel hammerhead deforms ever so slightly so the acceleration is continuous over a very small time scale.

To have speed jump instantaneously from 2 mph to 4 mph would require infinite acceleration. We never get infinite acceleration, so the answer is no, speed can not jump instantaneously.

Stephen T said:
Sorry if this is not worded well as I am unsure of how to, but I hope you understand and reply.
Smoothness means infinitely often differentiable, i.e. there are no vertices in the (position,time)-diagram and all of it differentials.
You're description by the use of instantaneous describes the continuity of this diagram, i.e. no gaps.
Although there might be no gaps, which would mean a division by zero, there still can be vertices, e.g. at the start when moving from zero to a constant acceleration.

There are two aspects of 'smoothness' assuming that you could apply an instant change in the force on an object, the velocity would not change instantly; the transition would be smooth. Also, an instantaneous change of velocity would not result in an instantaneous change in position - that would also involve a smooth change of distance.
It's worth while talking in terms of Calculus. An abrupt (discontinuous) change in a variable can result in an abrupt change in the rate of change in the (continuous) value another variable. Integrating a step change will produce a change in slope of the integral.

An instantaneous change in acceleration requires a force law with a discontinuity. In most cases, such force laws are only approximations. If you look closely enough you'll eventually hit the quantum world and discover that position is not only not always twice continuously differentiable. It is not even a well defined function of time.

In the kinematics of collisions, it is a good ruse to talk in terms of an Impulse. You ignore the fact that the collision has to have taken a finite time and you define Impulse as Force times time it's applied for. Given a 'before' situation, you can apply a given Impulse (any combination of forces times time) and produce the same result. If it's a truly elastic collision the same thing will happen to a steel ball bearing or a plastic 'superball' or even a very soft sponge (remembering that only an ideal sponge can be perfectly elastic)
It is very common to sidestep the problem of discontinuities in the Maths of Science.

jbriggs444

## 1. What is acceleration/deceleration?

Acceleration and deceleration are terms used to describe changes in an object's velocity. Acceleration refers to an increase in velocity, while deceleration refers to a decrease in velocity.

## 2. How is acceleration/deceleration measured?

Acceleration and deceleration can be measured using units of distance per time squared, such as meters per second squared (m/s^2). This measures the change in velocity over a certain amount of time.

## 3. Is acceleration/deceleration always smooth?

No, acceleration and deceleration can sometimes be abrupt or jerky, depending on the force or energy applied to the object. This can result in a non-smooth motion.

## 4. What factors affect the smoothness of acceleration/deceleration?

The smoothness of acceleration and deceleration can be affected by factors such as the amount of force applied, the mass of the object, and the surface on which the object is moving.

## 5. Can acceleration/deceleration be controlled?

Yes, acceleration and deceleration can be controlled by adjusting the amount of force applied to the object. For example, a car's acceleration can be controlled by how much pressure is applied to the gas pedal.

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