Problem 70 - College Physics

Question

If the time of flight of the ball is \(t\) seconds, at what point in time will the ball have zero vertical velocity? A. \(t / 4 \mathrm{s}\) B. \(t / 4 \mathrm{s}\) C. \(t / 2 \mathrm{s}\) D. \(t / 2 \mathrm{s}\) E. There is no place on the path where the vertical velocity is zero.

Step-by-Step Solution

Answer

C. \(t / 2 \) seconds.

Step 1: Understand the Problem

We have a ball in projectile motion, and we need to determine when during its flight the ball's vertical velocity will be zero. The total time of flight is denoted by \( t \).

Step 2: Recall Properties of Projectile Motion

In projectile motion, the vertical velocity of the ball is zero at the peak of its flight. This occurs halfway through the total time of flight because the path is symmetrical.

Step 3: Calculate the Time to Peak Height

Since the peak occurs halfway through the motion, we simply divide the total time of flight by 2. Thus, the time when the vertical velocity is zero is given by \( \frac{t}{2} \).

Key Concepts

Vertical Velocity

Vertical velocity in projectile motion is the component of the object's velocity that is directed along the vertical axis. When a ball is thrown into the air, it initially rises, slows down, stops momentarily at the peak of its trajectory, and then descends. This changing speed is due to the influence of gravity.

• Initial Vertical Velocity: When the ball is thrown upwards, it has a certain initial vertical velocity (\( v_{iy} \)).
• Deceleration: As the ball moves upwards, gravity decelerates it until it reaches the highest point where the vertical velocity is zero.
• Acceleration: After reaching the peak, gravity accelerates the ball downwards until it hits the ground or another surface.

The key takeaway is that the vertical velocity is only zero briefly at the peak of the projectile's path. At any other point in time during its flight, the ball has a non-zero vertical velocity as it moves up or down.

Time of Flight

Time of flight in projectile motion refers to the total duration for which the ball remains in the air. Understanding this concept is crucial as it helps determine other characteristics of the projectile's motion, such as when the vertical velocity reaches zero.

A few key points about the time of flight:

• The time of flight (\( t \)) is determined by both the initial velocity of the projectile and the angle of launch.
• In symmetrical projectile motion without air resistance, the time of ascent equals the time of descent.

To find specific moments during the flight, such as when the vertical velocity is zero, it is important to note that this happens exactly halfway through the time of flight. Hence, the time of flight provides a framework for understanding the entire motion.

Peak Height

Peak height represents the maximum vertical position a projectile reaches during its flight. At this point, the ball temporarily stops moving upwards before it starts to descend. This is directly related to vertical velocity and time of flight.

At peak height, the projectile's vertical velocity is zero, as the upward motion ceases and transitions to downward motion. This occurs because gravity has counteracted the initial upward force.

• The time to reach peak height in symmetrical projectile motion can be calculated using the formula: \( \text{Time to peak height} = \frac{t}{2} \), where \( t \) is the total time of flight.
• The peak height itself depends on the initial vertical velocity and can be influenced by factors such as launch angle and speed.

Understanding peak height is essential when analyzing projectile motion. It tells us not just about the maximum altitude reached but also provides insights into the symmetrical nature of projectile paths.