If a particle's acceleration is positive and its velocity is negative, is it speeding up?

Question

If the acceleration of a particle is positive and its velocity is negative, is the particle speeding up?

In other words, suppose that

• $y(t)$ is the position at $t$
• $y'(t)<0$
• $y'(t)$ is continuous and differentiable
• $y''(t)\geq0.$

Could the particle be speeding up?

The question originated here.

Answer 1

This is a question of semantics and of language usage rather than mathematics. People don’t usually use the phrase “speeding up” in everyday speech when an object’s velocity and acceleration have opposite signs. (In that case, the absolute value of the velocity will be decreasing.) Let’s say I’m playing with one of those toys consisting of a paddle and a ball attached by an elastic cord. As it goes downward, I’d naturally say that it slows down near the bottom of its motion, comes to a stop for a split second, and then speeds up as it starts to ascend.

But mathematically, its acceleration was nearly constant and directed upwards during its reversal of direction. So switching from “slowing down” to “speeding up” is a quirk of language.

To press the point, consider how it looks from another frame of reference. Let’s say I’m doing this in a glass elevator that’s ascending rapidly. Someone outside the elevator just focuses on the ball. (Maybe it glows and everything else is dark.) They see it speeding up the whole time.

This mismatch between everyday usage and mathematics is a broader phenomenon. Especially when it comes to negative numbers. People don’t say “I gained negative $500 on my investments last year”, or “my post got negative 5 likes”, but mathematically we have a single quantity with either positive or negative values.

Answer 2

Velocity is speed with a direction. I'm assuming when you say velocity <0, that means a direction is chosen and the particle is moving backwards along that direction while acceleration is adding forward speed to it. For example, imagine you're in a car facing uphill but rolling backwards down the hill. Now what happens when you push on the throttle pedal?

Answer 3

If a particle's acceleration is positive and its velocity is negative, is it speeding up?

A particle's velocity is in its direction of motion, while its acceleration is in the direction of the net force on it.

Therefore, a particle's acceleration and velocity being in opposite directions means that it must be retarding, that is, its speed must be decreasing, that is, it must be slowing down.

In your example, the particle is retarding while having a positive acceleration. If this feels surprising, remember that the frame of reference of this 1D motion can be legitimately flipped without changing the scenario, after which the particle will naturally still be retarding, but this time having a positive velocity and a negative acceleration.