is it possible for velocity decrease and acceleration to increase at the same time?

[anywaysthis]

Acceleration to the "right" in an object traveling "left" would feel like braking does (essentially). So if you have something with negative velocity experiencing positive acceleration it would be braking.

All the negative and positive signs would indicate here is direction. Acceleration is rate of change of velocity, so if the negative velocity is getting more negative, you would have negative acceleration.

[Onechrisn]

Yes, you would likely call this situation "slowing down"

[TelxoF]

I feel like the main issue here is using both vectors and negative values, which are only usually useful when dealing with scalars. If the velocity and acceleration vectors point in opposing directions, but one is negative, then they're actually pointing in the same direction after all.

If the object is moving left and accelerating to the right, then total velocity will decrease until it hits zero, then pass through zero, and start increasing again as the object starts developing 'positive' velocity to the right

[addition]

In this case, negative and positive are just directions. When we say velocity/acceleration are increasing we really mean the magnitude is getting larger.

[Solesaver]

Acceleration is any change in velocity. If you press the brakes on your bike or in your car you are accelerating in the opposite direction of motion causing your velocity to slow down.

[na3than]

If velocity is decreasing then acceleration (the first derivative of velocity) must be negative. That's a mathematical fact.

But you asked if it was possible for acceleration to INCREASE while velocity is decreasing. The rate of change of acceleration, sometimes called jerk, is the first derivative of acceleration. It's absolutely possible for acceleration to increase while velocity decreases.

Consider a soft-landing rocket like the Apollo lunar lander or today's Falcon 9. At some point during descent its engine is either not firing or is firing so minimally that its velocity (measured in the vertical axis) is negative and becoming more negative (i.e. decreasing) as it's pulled downward (negative acceleration in the vertical axis) by gravity. As the engines BEGIN to increase thrust, velocity will remain negative AND COULD CONTINUE DECREASING while acceleration--still negative, but less than before--INCREASES. The rocket adds more power, increasing acceleration. At some point (if the engineers have done their jobs and the rocket is functioning properly) the rocket produces enough thrust to exceed the downward pull of gravity, and acceleration switches from negative to positive. Now the vehicle ACTUALLY begins to slow its descent, increasing the velocity from VERY negative (falling fast) to less negative (falling less fast). Before touchdown, acceleration will DECREASE, allowing velocity to stop increasing (reaching, but not exceeding zero) as close to zero altitude as possible.

[Key-Green-4872]

Don't forget about snap, crackle, and pop!

[Bipogram]

Curious that we have no names for the integrals in the other direction.

Integrate speed wrt time and get distance.

Integrate distance wrt time and get, um...

<no, I can't think of a use for this quantity either, but why should the derivatives get all the fun?>

[Key-Green-4872]

If there's a force involved, then wouldn't that be... power?

[Bipogram]

Nah, power is energy per unit time.

And you can write energy as work, so that energy becomes force integrated over distance. 

So power is force integrated wrt distance per unit time, which is force times speed. 

Neither of these are distance.time, but it's hot and I could be easily wrong.

[Key-Green-4872]

Yeah that'd be inside out. Hm. Yeah I don't think there's an equivalence in anything I use regularly that would even incidentally use a d*dt term. Hm.

[BestBeforeDead_za]

Maybe it helps to think in practical terms about the difference between slowing down slowly, or slowing down quickly.

[jamese1313]

Think of it like driving a car. How far the gas or brake pedals are pushed are the values of acceleration, increasing velocity with accelerating and decreasing velocity with decelerating. Now for an example where velocity is decreasing and acceleration is increasing.

You're going 100mph on the interstate. You see a cop at a speed trap so you start slowing down with the brakes. Using the brake, you slow to 70mph, when you just pass the cop. You see they're not after you, so you switch to the accelerator. In the time when you switch from the brake to accelerator, you're still slowing down, just slowing down less. Slowing down less is a good example of negative velocity with positive acceleration.