Problem 77 - Physics for Scientists and Engineers with Modern Physics

Question

Romeo is chucking pebbles gently up to Juliet's window, and he wants the pebbles to hit the window with only a horizontal component of velocity. He is standing at the edge of a rose garden \(8.0 \mathrm{~m}\) below her window and \(9.0 \mathrm{~m}\) from the base of the wall (Fig. 3-55). How fast are the pebbles going when they hit her window?

Step-by-Step Solution

Answer

Pebbles hit the window with a speed of approximately 3.52 m/s horizontally.

Step 1: Understand the Scenario

Romeo is throwing pebbles at an angle such that they will hit Juliet's window with only horizontal velocity at the moment of impact. This means that the vertical velocity component at the window must be zero. Therefore, at the top of the path, where the pebble's vertical rise equals the height difference, pebbles stop moving upwards and only have horizontal speed.

Step 2: Use Kinematics for Vertical Motion

The window is 8.0 m above the point where Romeo is throwing the pebbles. From kinematics, the vertical motion of the pebble is given by: \[ v_y^2 = v_{y0}^2 - 2g h \]Where:- \( v_y \) is the final vertical velocity (0 m/s at the window)- \( v_{y0} \) is the initial vertical velocity- \( g \) is the acceleration due to gravity (9.8 m/s²)- \( h \) is the height (8.0 m)Setting \( v_y = 0 \) gives: \[ 0 = v_{y0}^2 - 2(9.8)(8.0) \]

Step 3: Solve for Initial Vertical Velocity

Rearrange the equation from Step 2 to find \( v_{y0} \):\[ v_{y0}^2 = 2 \times 9.8 \times 8.0 = 156.8 \]Thus, the initial vertical velocity is:\[ v_{y0} = \sqrt{156.8} \approx 12.52 \text{ m/s} \]

Step 4: Determine Horizontal Velocity

Since the pebbles hit the window only with horizontal velocity, \( v_{x} \) remains constant throughout the motion. The horizontal distance traveled, \( d = 9.0 \text{ m} \), and the time of flight, \( t \), gives:\[ t = \frac{2v_{y0}}{g} = \frac{2 \times 12.52}{9.8} \approx 2.56 \text{ s} \]Using horizontal motion: \[ d = v_{x} \times t \]\[ 9.0 = v_{x} \times 2.56 \]

Step 5: Solve for Horizontal Velocity

Rearrange to solve for \( v_{x} \):\[ v_{x} = \frac{9.0}{2.56} \approx 3.52 \text{ m/s} \]Since the pebbles hit the window with only horizontal velocity, this is also the speed when they hit the window.

Key Concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. This scientific field plays a crucial role in understanding projectile motion. In the exercise, we focus on different aspects of the pebble's trajectory as it moves towards Juliet's window.

When exploring kinematics in projectile motion, we typically analyze both vertical and horizontal components. These components can be studied separately because they operate independently of each other but are part of the same motion.

By applying kinematic equations like the one used in this exercise:

• Vertical motion: \( v_y^2 = v_{y0}^2 - 2gh \)
• Horizontal motion: \( d = v_x \times t \)

we can effectively solve for factors such as time, distance, and velocity of the projectile. Understanding kinematics helps us deconstruct motion into manageable parts and facilitates solving complex motion problems.

Horizontal Velocity

Horizontal velocity in projectile motion refers to the component of velocity that acts along the horizontal axis. In our scenario, Romeo wants the pebbles to hit the window with only horizontal velocity.

This means, by the time they reach the window, their vertical speed should be zero, and only the horizontal speed is significant. The horizontal velocity remains constant throughout the flight of the pebble due to the absence of horizontal acceleration (neglecting air resistance).

In the exercise, the constant horizontal velocity is calculated using the formula:
\[d = v_x \times t\]

• where \(d\) is the distance travelled horizontally, \(v_x\) is horizontal velocity, and \(t\) is the time of flight.

This concept helps us understand how objects move over flat surfaces or reach specified horizontal distances.

Vertical Velocity

Vertical velocity is the component of velocity that acts along the vertical axis, influenced by gravity. In the context of this exercise, the vertical velocity of the pebbles needs careful consideration.

Initially, the pebbles have an upward vertical velocity that is counteracted by gravity over time. As explained in the vertical motion equation \( v_y^2 = v_{y0}^2 - 2gh \), the vertical velocity decreases until it reaches zero at the window, indicating that the pebble has stopped ascending.

Using vertical kinematic equations, we can determine the initial vertical velocity, crucial for calculating time of flight. This helps ensure the pebbles meet the desired condition of hitting the window with no vertical speed and only horizontal velocity.