Problem 6 - Pearson Physics

Question

A cannon \(95 \mathrm{~m}\) away shoots a cannonball straight up in the air with an initial speed of \(44 \mathrm{~m} / \mathrm{s}\). What is the speed of the cannonball when you hear the shot?

Step-by-Step Solution

Answer

The speed of the cannonball when you hear the shot is approximately 41.29 m/s.

Step 1: Identify the Known Variables

First, identify the given variables in the problem. We know that the distance from the cannon to the observer is \(95\, \text{m}\), the initial speed of the cannonball is \(44\, \text{m/s}\), and the speed of sound is approximately \(343\, \text{m/s}\) in air at room temperature.

Step 2: Calculate Time for Sound to Travel

Calculate the time it takes for the sound of the cannonball being fired to reach you. Using the formula for time, \( t = \frac{d}{v} \), where \( d = 95 \text{ m} \) is the distance and \( v = 343 \text{ m/s} \) is the speed of sound.\[t = \frac{95 \text{ m}}{343 \text{ m/s}} \approx 0.277 \text{ s}\]

Step 3: Determine Time for Cannonball to Rise

In the time it takes for the sound to reach you (\(0.277\, \text{s}\)), calculate how much the cannonball has slowed due to gravity. The formula for the final velocity when shot upwards, taking gravity \(g = 9.8 \text{ m/s}^2\), is \( v = u - g t \), where \( u = 44 \text{ m/s} \).\[v = 44 \text{ m/s} - 9.8 \text{ m/s}^2 \times 0.277 \text{ s} \approx 41.29 \text{ m/s}\]

Step 4: Conclude the Speed of the Cannonball

The speed of the cannonball when you hear the shot is \(41.29\, \text{m/s}\), considering that the effect of gravity would have slowed it down from its initial speed of \(44\, \text{m/s}\) during the time it took for the sound to reach your ears.

Key Concepts

Projectile motion

Projectile motion occurs when an object is thrown or propelled into the air and is influenced by gravity. It involves two components of motion: horizontal and vertical. In many problems, like the cannon shot example, the object is launched in a straight vertical plane. This simplifies calculations since only gravity affects the vertical motion; the horizontal speed, if any, remains constant.

When shooting an object directly up, there are important points to consider:

• Initial velocity: The speed at which the object is launched. For the cannonball, this is 44 m/s.
• Acceleration due to gravity: Gravity pulls the object downward at approximately 9.8 m/s². This deceleration affects the upward motion.
• Final velocity: The speed at a specific time after launch, influenced by gravity over time.

Understanding these components helps us calculate the motion of the projectile effectively.

Speed of sound

The speed of sound is the rate at which sound waves travel through a medium. In air, sound travels at about 343 m/s at room temperature. This velocity can vary based on factors such as air pressure and temperature.

When a cannon fires, the sound waves travel from the cannon to the observer. To figure out how long it takes these waves to reach someone standing some distance away, we use the formula:

• \( t = \frac{d}{v} \)

where \( d = 95 \, \text{m} \,\) is the distance and \( v = 343 \, \text{m/s} \, \) is the speed of sound. In our example, it takes approximately 0.277 seconds for the sound to travel 95 meters. This delay in hearing the shot is crucial for timing and understanding projectile motion.

Kinematics

Kinematics is the study of the motion of objects without considering the forces that cause the motion. It involves measurements like velocity, acceleration, displacement, and time.

For the cannonball, we consider its vertical flight. The key kinematic equation used here is:

• \( v = u - g t \)

where \( u \, \) is the initial velocity (44 m/s for the cannonball), \( g \, = \, 9.8 \, \text{m/s}^2 \, \) is the acceleration due to gravity, and \( t = 0.277 \, \text{s} \, \) is the time elapsed when you hear the shot.

This formula helps us calculate that the cannonball's speed at the moment you hear the sound of the shot is approximately 41.29 m/s, indicating how much it has slowed down due to gravity.

Gravity

Gravity is the force pulling objects toward the Earth, with an acceleration of approximately 9.8 m/s². It plays a critical role in projectile motion, affecting the object from the moment it is launched.

In our problem, gravity acts on the cannonball as soon as it is fired. As it ascends, gravity slows it down at a rate of 9.8 m/s², reducing its upward speed. This means that the longer the cannonball is in the air, the slower it moves upwards until it eventually stops and begins descending.
Gravity's influence is constant and acts in the opposite direction of the projectile's motion, making it essential to account for it in kinematic equations. Without this force, a projectile would continue moving upwards indefinitely, defying the natural laws of physics.