# Effect on speed when decreasing the magnitude of acceleration

I'm struggling to come up with an answer to the following situation and question:

"Suppose you are driving your car along a straight road in the positive direction and are speeding up (increasing your velocity in the positive direction). You now relax slowly on the gas pedal, decreasing the magnitude of your acceleration (you do not use the brake).

Are you increasing or decreasing your speed while the magnitude of the acceleration decreases?"

I know that decreasing the magnitude of acceleration doesn't necessarily mean an object is decelerating (e.g. a car accelerating from 0-50mph, and then again from 50-60mph in the same amount of time - the magnitude of the acceleration is smaller but the car is still accelerating), but in the above case, I struggle to see how slowly taking your foot off of the gas could potentially increase speed, even though your foot is still on the accelerator. My gut is telling me that the car's speed would decrease, but I'm having trouble justifying that and don't even know if that is correct.

I would love to hear what you all think.

Thank you.

Hopefully this helps. As you can see, acceleration is a change in velocity. When you're decreasing acceleration, you're decreasing the change in velocity. When the acceleration hits zero, your velocity is constant.

The car situation might seem tricky because of wind resistance, friction, in real life your car will slow down once you take your foot off the gas pedal However, in a system without friction and wind resistance because of inertia, it would keep moving at the same velocity if a = 0 as seen above and increasing in speed when a > 0.

Acceleration is the rate of change of speed: $$a = \frac{d v}{d t}$$. As long as it is positive, the change of speed is positive.

Taking the foot slowly off the accelerator pedal means that $$\frac{d a}{d t} < 0$$, but if we pretend the wind and friction are absent, you will eventually end up with $$a = 0$$ (this is just how cars work, you need to use brakes to get $$a < 0$$ in these idealized conditions). And $$a = \frac{d v}{d t} = 0$$ means constant speed.

So, in summary, while the magnitude of the acceleration decreases (but does not reach zero), the speed will increase in smaller and smaller increments.

Acceleration is the rate change of velocity. This is all you need to know.

But your doubt probably stems from not understanding what rate change of velocity means.

What this means is by how much would an object's velocity increase or decrease in a finite amount of time. From this, it should be evident that a jump from 50 to 60 mph should happen even if the acceleration is now smaller than the previous acceleration taking the car from 0 to 50 mph.