Yes, it is possible to have positive acceleration without having positive velocity, and vice versa. Let's consider each case separately:
Positive acceleration without positive velocity: This situation can occur when an object is slowing down while moving in the positive direction. For example, imagine a car traveling in the positive direction at a high speed. If the driver applies the brakes, the car will experience a negative acceleration (deceleration), but its velocity will remain positive until it comes to a stop. During this time, the car has a positive acceleration (in the negative direction) but a positive velocity.
Positive velocity without positive acceleration: This scenario arises when an object is moving at a constant speed in the positive direction, or when its velocity is positive but not changing. For instance, consider a ball rolling horizontally across a frictionless surface with no external forces acting on it. As long as the ball maintains a constant speed, its velocity remains positive, but there is no acceleration because acceleration is defined as a change in velocity over time.
In summary, positive acceleration without positive velocity and positive velocity without positive acceleration are possible because velocity and acceleration are independent quantities. Velocity refers to the speed and direction of an object's motion, while acceleration describes how quickly the velocity is changing. They can have different signs and can exist independently of each other in various motion scenarios.