An object is thrown upward at a certain angle above the ground, eventually returning to erth.1.Is there any place along the trajectory where the velocity and acceleration are perpendicular? If so, where? 2. Is there any place where the velocity and acceleration are parallel? if so, where? in each case, explain.

Yes and no. Acceleration is always down (vertical). The velocity is horizontal at the top of the trajectory, and is never down unless the object is thrown vertically.

To determine if there is any place along the trajectory where the velocity and acceleration of the object are perpendicular or parallel, we need to understand the relationship between velocity, acceleration, and the direction of motion.

1. Place where velocity and acceleration are perpendicular:
Velocity is a vector quantity that indicates the object's speed and direction of motion, while acceleration is also a vector quantity that represents the rate of change of velocity. The direction of velocity and acceleration can be represented as arrows. When two vectors are perpendicular, the angle between them is 90 degrees.

In the case of an object thrown upward, as it ascends, its velocity decreases. At the peak of its trajectory, the object momentarily stops before descending. At this point, the velocity changes direction and starts to decrease in the opposite direction. Meanwhile, the acceleration due to gravity acts vertically downward throughout the motion.

At the exact moment when the object reaches its maximum height and starts to descend, the velocity is momentarily zero. This is the point where the velocity and acceleration are perpendicular. Since the acceleration due to gravity is acting straight downward and the velocity is momentarily zero, the vectors representing velocity and acceleration are at a 90-degree angle. It occurs at the highest point of the object's trajectory.

2. Place where velocity and acceleration are parallel:
For the velocity and acceleration to be parallel, both vectors must have the same direction. In the case of the object thrown upward, the velocity is initially in the upward direction, while the acceleration due to gravity is in the downward direction throughout the motion.

While the object is going upward or coming downward, the velocity and acceleration are always in opposite directions. Therefore, there is no place along the object's trajectory where the velocity and acceleration are parallel.

In conclusion:
1. The place along the trajectory where the velocity and acceleration are perpendicular is at the highest point of the object's motion, where the velocity momentarily becomes zero as the object changes direction from upward to downward.
2. There is no place along the trajectory where the velocity and acceleration are parallel due to their opposite directions during the object's motion.