Question

A baseball player hits a high fly ball to the outfield. (a) Is there a point during the flight of the ball where its velocity is parallel to its acceleration? (b) Is there a point where the ball's velocity is perpendicular to its acceleration? Explain in each case.

Short answer

(a) Yes, when descending after the peak. (b) Yes, at the peak.

Step-by-step solution

Define the Scenario

In projectile motion, such as a fly ball hit in the air, the only acceleration acting on the ball after it is hit is due to gravity. Gravity acts downwards, so the acceleration vector points down towards the ground.

Analyze Velocity and Acceleration Relationship

The velocity of the ball has both a horizontal and vertical component. Initially, it has an upward and forward velocity. As the ball rises, its vertical velocity decreases due to the constant downward acceleration (gravity). At the peak, the vertical component of the velocity is zero.

Identify Parallel Velocity and Acceleration

At the peak of the trajectory, just as the ball starts descending, the horizontal component is non-zero and vertical velocity is zero. But just after the peak, the ball's velocity direction changes downward, aligning parallel to acceleration (gravity), when it starts descending.

Identify Perpendicular Velocity and Acceleration

Initially, the ball has both upward and forward velocity, with acceleration acting downward. At the peak of its flight, the vertical velocity becomes zero, leaving only horizontal velocity. At this instant, the velocity is perpendicular to acceleration because velocity is horizontal (parallel to ground) and acceleration is vertical (perpendicular to ground).

Key concepts

Velocity and Acceleration

In the context of projectile motion, velocity and acceleration have distinct behaviors that define the flight path of objects like fly balls. Velocity is a vector quantity, meaning it has both magnitude and direction. It has two components in projectile motion: horizontal and vertical.

• The *horizontal velocity* is constant, as there are no horizontal forces acting once the ball is hit, ignoring air resistance.
• The *vertical velocity* changes, as it is influenced by gravity.

Acceleration, in projectile motion, is due to gravity. Its fixed direction is downward, toward the center of the Earth, with a magnitude of approximately 9.81 m/s². Importantly, this means that throughout the ball’s flight, its acceleration is constant in magnitude and direction.

During the ball's ascent, the vertical velocity component decreases as gravity opposes it. Once the ball reaches its highest point, the vertical velocity becomes zero, and then gravity causes it to increase in the downward direction as the ball descends. This interplay between velocity and acceleration is key to understanding projectile motion.

Gravity's Impact on Motion

Gravity plays a crucial role in determining the trajectory of the baseball once it has been hit. It is the constant force causing objects to accelerate downward at 9.81 m/s². In projectile motion, gravity doesn't just act on objects for a moment; it continuously influences every moment of the projectile's flight.

• **Ascending phase**: Including the time immediately after the ball is hit until it reaches its peak, gravity decreases the ball’s upward velocity.
• **Peak point**: The vertical component of the velocity is zero at the peak. Gravity’s influence is solely vertical at this point.
• **Descending phase**: As the ball begins to fall, gravity increases the ball’s vertical velocity in the downward direction.

Gravity's continuous downward pull is the reason why after being struck, even a baseball hit at an angle follows a curved path instead of traveling straight. Without gravity, a hit ball would continue in a straight line rather than arcing towards the ground.

Trajectory Analysis

Analyzing the trajectory of a projectile like a baseball allows us to understand the link between its path and the forces at play. The trajectory is the curved path that the baseball follows due to the combination of its initial velocity and the acceleration due to gravity.

• **At launch**: Both horizontal and vertical velocity components are present, with the ball describing an upward and forward path.
• **At the peak**: The vertical velocity is zero, and the trajectory begins to change direction, marking the transition from ascent to descent.
• **During descent**: The ball accelerates downward under gravity’s influence, combining with the horizontal motion to complete the parabolic path.

The trajectory can be analyzed using physics equations to determine key attributes such as the time of flight, maximum height, and range of the projectile. Understanding the trajectory forms a foundation for solving problems involving projectile motion, as it integrates the concepts of velocity, acceleration, and the effects of gravity over time.