An automobile is traveling west. Can it have a westward velocity and at the same time, an eastward acceleration? If so, under what circumstances?

Velocity is the ratio of displacement to the time taken. Acceleration is the rate of change of velocity and can be positive (known as acceleration) or negative (known as deceleration). It is entirely possible for an automobile to have a velocity in the west direction and motion in the west direction, yet (at the same time) an acceleration in the east direction. This is possible when we are decelerating or applying the brakes on the vehicle. When that happens, the automobile will feel an acceleration in the opposite direction, which is east in this case. Ultimately, the vehicle will come to a stop, but until that instant, it will have velocity and motion in the west direction while the acceleration will be in the east direction.

One can compare this to a ball thrown upwards (away from Earth). The velocity and motion is in the upwards direction (at least for a while), while the acceleration (due to gravity) is working in the opposite direction (towards Earth). Ultimately the ball will stop going up, fall down and come to a rest.

Hope this helps.