Question

In ideal projectile motion, the velocity and acceleration of the projectile at its maximum height are, respectively, a) horizontal, vertical c) zero, zero. downward. d) zero, vertical downward. b) horizontal, zero. e) zero, horizontal.

Short answer

Answer: At its maximum height, the projectile's velocity is horizontal and its acceleration is vertical downward.

Step-by-step solution

Recall the main concepts of projectile motion

In ideal projectile motion, a projectile is launched with an initial velocity that has both vertical and horizontal components. The projectile's motion is affected only by gravity, which acts vertically downward.

Analyze projectile velocity at the maximum height

At the maximum height of the projectile, the vertical velocity component becomes zero because the projectile momentarily stops moving upwards and starts falling back to the ground. However, the horizontal component of velocity remains constant and unaffected by gravity. Therefore, the projectile's velocity at its maximum height is horizontal.

Analyze projectile acceleration at the maximum height

The acceleration of the projectile is due to gravity, which always acts vertically downwards. This acceleration remains constant throughout the projectile's motion, including when it's at its maximum height. Therefore, the projectile's acceleration at its maximum height is vertical downward.

Identify the correct option

Based on the analysis above, the correct statement about the projectile's velocity and acceleration at its maximum height is: - Velocity: horizontal - Acceleration: vertical downward The correct option is (a).

Key concepts

Velocity Components

Understanding the concepts of velocity components is vital when dealing with projectile motion. In essence, any velocity can be split into two components: horizontal and vertical. These components help us to analyze a projectile’s path.

• **Horizontal Component**: This remains constant throughout the motion, assuming there is no air resistance.
• **Vertical Component**: This changes over time due to the influence of gravity.

Breaking velocity into these components allows us to understand how the projectile moves. At the start, it has both a vertical and a horizontal motion. As it rises to the peak of its path, the vertical velocity decreases while the horizontal component stays the same.

Maximum Height

The concept of maximum height is crucial in projectile motion. It is the highest point a projectile can reach. Just for a brief moment at this point, the projectile's vertical velocity component ceases to exist. This happens because gravity pulls the projectile downwards, slowing the vertical motion to zero before reversing its direction. At this point, only the horizontal component of velocity remains, as gravity doesn’t affect it.

Gravitational Acceleration

Gravitational acceleration is a force that acts on every object with mass. It plays a pivotal role in a projectile's motion, influencing the vertical component of velocity but not affecting the horizontal component.

• Constant Force: Gravity always acts downward, uniformly accelerating any downward motion.
• Value: On Earth, gravitational acceleration is approximately 9.81 m/s².

This force causes the projectile to reach its maximum height and eventually return to the ground, ensuring that projectile motion is always downward in the vertical axis.

Horizontal Velocity

Horizontal velocity represents the constant motion of a projectile across a plane. As there is typically no opposing force like air resistance considered in ideal projectile motion, the horizontal velocity stays constant throughout the entire flight. This means regardless of the changes happening vertically, horizontally, the object maintains its speed unless otherwise acted upon. This consistent velocity helps in predicting how far a projectile can travel along the horizontal plane.

Vertical Velocity

Vertical velocity is ever-changing due to gravitational forces affecting the projectile motion. Unlike the horizontal component, this component is distributed in a parabolic motion.

• **At Launch**: It starts at its maximum value and decreases as height increases.
• **At Maximum Height**: The vertical velocity becomes zero at the peak of motion.
• **On Descent**: Gravity accelerates this velocity in the downward direction, increasing its magnitude until it reaches the ground.

The vertical velocity component is crucial to understanding the rise and fall dynamics of a projectile's motion.