Acceleration of Moving Particles: Explained

[Huzaifa]

I am not able to understand the following paragraph from my Physics textbook;

The acceleration of a moving particle may be positive or negative. If the speed of particle is increasing with time then acceleration is positive and if the speed is decreasing with time then acceleration is negative.
For positive acceleration the velocity vector and acceleration vector are in the same direction. But for negative acceleration, the velocity and acceleration vectors are opposite.

 

[Ibix]

What bits do make sense to you? Do you understand what a vector is, for example?

I have to say it's not the best description of its subject I've read.

 

[Mister T]

I am not able to understand the following paragraph from my Physics textbook;

The acceleration of a moving particle may be positive or negative. If the speed of particle is increasing with time then acceleration is positive and if the speed is decreasing with time then acceleration is negative.

This statement is true only for objects moving in the positive direction (the direction of increasing position). If an object is moving with decreasing position and is speeding up then the acceleration is negative. And if it's slowing down the acceleration is positive.

You can see this for yourself by making a velocity-time graph of an objects motion. If the slope of the graph is positive the acceleration is positive, and if the slope is negative the acceleration is negative.

The above quoted passage from your textbook can be made correct if you replace "speed" with "velocity".

For positive acceleration the velocity vector and acceleration vector are in the same direction. But for negative acceleration, the velocity and acceleration vectors are opposite.

This statement is valid.

Edit: Sorry. This statement is also incorrect. It should read that if the velocity and acceleration vectors point in the same direction the particle is speeding up, if they point in opposite directions the particle is slowing down.

 

[mjc123]

That statement is not valid. For positive acceleration, the acceleration vector points in the positive direction. This is perfectly compatible with the velocity vector pointing in the negative direction, but decreasing in magnitude.

Consider a particle in harmonic motion along the x-axis about the origin, with x = x₀sinωt.
During the first quarter of the cycle, displacement is positive, velocity is positive, acceleration is negative and speed is decreasing.
2nd quarter: displacement +ve, velocity -ve, acceleration -ve, speed increasing.
3rd quarter: displacement -ve, velocity -ve, acceleration +ve, speed decreasing.
4th quarter: displacement -ve, velocity +ve, acceleration +ve, speed increasing.
Note there is no correlation between the signs of velocity and acceleration, or increasing/decreasing speed.

 

[PeroK]

I am not able to understand the following paragraph from my Physics textbook;

The paragraph is nonsense. To begin with it assumes one dimensional motion. In general, velocity and acceleration are one, two or three dimensional vectors.

Even for one dimensional motion the paragraph is hopelessly wrong. Take the example of gravitational acceleration. If we take down to be the positive direction, then the acceleration is $g = +9.8 m/s$. This acceleration is the same regardless of whether an object is moving upwards and hence slowing down, or the object is falling and hence speeding up.

In other words, the sign of the acceleration is determined by the positive axis and not the motion of the particle.

PS I think we've seen this question before on this site.

 

[vanhees71]

It is well known that teaching mechanics without first introducing vectors leads to trouble. One should avoid textbooks not making the kinematical concepts clear by first introducing vectors. This little effort is well justified, because (a) you need vectors and tensors for all of physics, (b) the most intuitive examples of vectors directly related to everyday experience occur in mechanics (both point-particle and continuum mechanics!), and (c) it avoids confusing many-words explanations saving students a lot of time trying to understand ill-defined concepts.

As of any medicine also an overdose of well meant didactics causes more damage than help!

 

[jbriggs444]

The writer of the paragraph in the original post appears to be adopting a coordinate system whose axes are tied to the current motion of the object and casually ignoring the possibility of an object at rest or of acceleration on anything other than a straight-line track.

This is not an ideal thing to be doing. It undermines the ability to then teach about frames of reference. Pick something easier to start with -- inertial frames of reference and non-rotating coordinate systems.

That said, think of it like a car that is moving forward on a straight track. Hit the gas and you have positive acceleration. Hit the brakes and you have negative acceleration (deceleration).

If the car is moving backward, shift into reverse and the same still holds. Hit the gas and it goes faster in reverse - positive acceleration in the direction of motion. Hit the brakes and it goes slower in reverse - negative acceleration in the direction of motion.

For a car at rest, just label the front as positive if there is an identifiable front. For a particle at rest there will not be an identifiable front. So you can label the direction of the acceleration as positive. Or flip a coin.

In my opinion, it would be better to tie your coordinate system to the road instead of the car. Put the zero marker where you start and the end marker at some positive coordinate where you expect to end up. Then positive acceleration gets you to your destination faster and negative acceleration prevents it.