Can an object have a velocity that is positive & increasing while its acceleration is decreaseing? Explain.?

Question

My physics teacher put this up on the forum for fun discussion. Here's my reply

No. As long as by acceleration you mean total acceleration in the vector. (I went on to explain using gravity and a rocket, but that won't fit in here)

He replied with

Think about a v vs. t graph that is not a straight line (recall that the slope of this graph represents the acceleration). Suppose from t= 0s to t= 3s the line is roughly straight and has a slope of 4. Then, from t= 3s to t= 6s, the line is again roughly straight but has a different slope of 2. Is the speed still increasing after t= 3s?

I say, yes the speed is increasing after t=3s (a=2) but what about at t=3s? there is obviously a negative acceleration there, but at that point surely the velocity is not still increasing because for the object to slow it's total acceleration must be negative. What do you think?

Greg

Answer 1

at t=3s the acceleration is decreasing but isn't negative... if it were negative then the line would be sloping downward.

to answer your primary question, yes. if the acceleration decreases from, say, 4 m/x^2 to 2 m/s^2, the velocity is obviously still increasing either way (as long as the acceleration stays positive).

Answer 2

Greg, yes it can. The object will have a velocity that is positive and increasing so long as the acceleration is a positive value. In graphic form you described as long as the slope is >0 the velocity is increasing.

For example, say that an object is moving at 10 m/sec and accelerating at a rate of 2m/s^2 (2m/s/s). The rate of acceleration is deceasing at a rate of .25m/s^3(.25m/s/s/s). It would take 8 sec to reach constant acceleration, and at that time the object would be at its maximum acceleration. Thereafter the acceleration would start to fall until it reached zero and then negative acceleration. When the acceleration is zero the velocity is constant and not increasing. When the acceleration becomes negative the velocity decreases.

Answer 3

Acceleration is the slope of a velocity time graph. If it is positive, then yes, velocity is increasing but in the sense that if it is positive it becomes more positive or if it is negative, it is becoming less negative( meaning it is slowing down as a speed). Slowing from -3 m/s to -2 m/s is a positive acceleration and the velocity is increasing in the positive direction ( becoming less negative). In circular motion the acceleration is constantly changing (NOT constant) because the directioin is changing, even though the magnitude of the acceleration might be constant. v and a are at right angles in circular motion. but again when a is positive, v is increasing ( becoming less negative)

Answer 4

Alright, how about this. Let's look at another v vs. t graph.

Instead of a series of fairly straight lines, let's smooth these lines out to create a curve where the velocity keeps going up, but begins to flatten out. Similar to the one described by your teacher, but let's have the velocity approaching some upper limit, an "asymptote". The velocity is constantly increasing, but never reaches this asymptote (see link - let's examine the upper half of this graph, with the vertical axis being velocity and the horizontal, time).

Now, since the velocity curve is approaching this "asymptotic" line, but never reaches it, it begins to flatten out. Therefore, the acceleration is decreasing. It isn't negative, and is still positive, but it decreases in magnitude as the velocity approaches the upper limit. The rate of change of velocity - i.e. the acceleration - is decreasing.

Answer 5

of course it can.

say you're traveling at 10 m/s. Say your acceleration is 1 m/s^2, decreasing at 0.1 m/s^2, per second.

if we approximate very roughly i.e. go by steps:
- in the first second you'll be seeing an acceleration of 1 m/s^2, so you'll reach a speed of 11m/s.
- in the 2nd second, you'll be seeing an acceleration of just 0.9 m/s^2. So your speed will increase to 11.9 m/s
- and so on (for a while)

so you'll have a positive, increasing speed, with a positive, but decreasing, acceleration.

of course once the acceleration has reached zero, it will become negative, and then your speed will start decreasing

Answer 6

What if the velocity is indeed increasing, but increasing at a slower rate each time, doesn't this mean that its acceleration is decreasing?? That means the object is decelerating, doesn't it?

5 initially---> 10---->14---->17---->19---->20---->20--->?
Eventually the value becomes constant. Acceleration is decreasing. What do you think??

Answer 7

Think of this analogy. You driving along in a car at 100kms an hour and you see a 50 buck note on the side of the road. You hit the brakes. In this case your acceleration is negative ie slowing(decreasing) but your velocity is still positive as you are moving forward. When you reverse back to the 50 buck note your acceleration becomes positive as you are increasing your speed but your velocity becomes negative as you are going backwards.P.s that fifty was mine i'd like it back please

Answer 8

Sure, Acceleration is rate of change of speed. But the acceleration can be changing too. I don't know what you call the rate of change of acceleration, but it is the rate of change of the rate of change of speed. It is the second derivative of velocity and the third derivative of position with time.

For example, a sky diver jumps out of an airplane and starts falling. After 1 second he is falling at 32 feet per second. After 2 seconds he is falling at 64 feet per second. This is because the acceleration of gravity is 32 feet per second per second. But after he gets going fast enough, air friction comes into play. So maybe in the 3rd second he is only falling at 90 feet per second and not 96 as you would expect in a vacuum. His downward velocity is still increasing, but it is increasing more and more slowly as time goes on. Eventually, he reaches terminal velocity, and his acceleration is then zero. But for that first few seconds he was falling with increasing velocity and decreasing acceleration.

Answer 9

In Physics acceleration (symbol: a) is defined as the rate of change (or derivative with respect to time) of velocity. It is thus a vector quantity with dimension length/time².

To accelerate an object is to change its velocity (or direction — like in case of uniform circular motion) in relation to time. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and the instantaneous acceleration of an objection is given by the equation
a=dV/dt,
where a is the acceleration vector (as acceleration is a vector, it must be described with both a direction and a magnitude). v is the velocity and t is time.

As you can clearly see that since Acceleration is directly proportional to velocity it will not have acceleration [it may have deceleration which is negative].

Cheers!

Answer 10

Yes I think a pendulum illustrates this. Check out my reasoning...

On release at an angle the pendulum has a steadily increasing velocity until it reaches it's maximum velocity as it swings through the vertical. Whilst it does this it's acceleration is decreasing because it is subject to frictional forces.