Can an object reverse its direction of acceleration even though it continues to move in the same direction?

Get on a car, turn it on and press the accelerator! (${{a}}>0$)... then press the break (${{a}}<0$). Until the car is steady, the situation is as you described.

Absolutely.

Acceleration is the change in velocity, so when you say that the acceleration reverses in direction, it means that the object is transferring from either speeding up to slowing down, like a skateboard which just passed the bottom of a U-shaped ramp, or from slowing down to speeding up. Think of the skateboard as having just crested a hill. It was previously slowing down as it climbed the hill, then it reached the top, where its acceleration was 0, and then it began descending the hill, so its forward acceleration became positive.

Of course, this is a very general example for a very general question. You can apply this idea to a car that has just accelerated and is now applying the brakes (positive to negative acceleration), or a person falling onto a trampoline (negative to positive).

Of course. Acceleration is the rate of change of the velocity.

For motion in a line, if the object is slowing down, the acceleration is opposite the velocity. If the object is speeding up, the acceleration is in the direction of the velocity.

Imagine you're pedalling a bike, gaining more and more speed and then, suddenly, stop pedalling and apply the brakes. Your motion is still in the same direction but you suddenly change from speeding up to slowing down - your acceleration suddenly changes direction but you continue to move in the same direction.

A good example to study to get an intuitive feeling for the relationship between velocity and acceleration is the back and forth motion of a mass attached to a spring.

Try the application at this link which not only animates the motion but also plots the position, velocity, and acceleration. When the acceleration has the opposite sign of the velocity, the acceleration is in the opposite direction of the velocity.

(Temporarily pretend you are on the second floor of a building.) Jump up and down in one place. After you leave the floor:

• your displacement is always positive (i.e., above the floor);
• your velocity is positive (rising), zero (at your highest displacement), then negative (falling back to the floor); and
• your acceleration is constant and negative -- it's gravity steering you back to the floor.

If you walk downstairs to the first floor, your displacement is always negative (unless you are capable of jumping as high as the distance between floors, you are always below where you started on the second floor), your velocity and acceleration, however follow the same pattern.

If alternatively, the floor fails and you fall to the first floor, you displacement will be initially positive and then negative.

So even though in this very simple experiment your acceleration is always negative (down), your velocity and displacement can be positive, zero, or negative in any combination.

For more examples --

• A bullet flying through the air has large velocity, a small drag acceleration directed (ideally) opposite to its velocity, and a smaller gravitational acceleration pointed down. It takes a long time for the drag to overcome the initial velocity of the bullet. For bullets fired horizontally near the surface of the Earth, the gravitational force will bring the bullet to the ground before drag will arrest its forward motion.
• A beginning roller skater (when upright and rolling) is always experiencing a little drag acceleration opposite their velocity. Bearing aren't perfect and the gummy residue that seems to collect on wheels and rinks resists the rolling of the wheels (by not letting go when the wheel tries to roll on).
• Large ships -- you can throw it in reverse, but it's going to take a while to get all that mass to (first) stop and (then) start moving back the other way.
• During landing, there are several components applying acceleration to the rear of the plane -- for instance, the engines, having been put in reverse, which just a few minutes ago were providing enough forward thrust to keep the plane in the air, take several seconds to erase the forward velocity of the plane.

Get in a car and step on the accelerator pedal. That causes the car to accelerate forward. Now slam on the brakes. This causes the acceleration to point in the opposite of the direction of motion even though the car is still moving forward.