Problem 77 - Physics for Scientists and Engineers with Modern Physics

Question

Romeo is chucking pebbles gently up to Julict'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 below her window and 9.0 \(\mathrm{m}\) from the base of the wall (Fig. 55 ). How fast are the pebbles going when they hit her window?

Step-by-Step Solution

Answer

The pebbles hit the window at 7.03 m/s horizontally.

Step 1: Understand Problem Requirements

We are given the vertical height of 8.0 m from Romeo to Juliet's window and the horizontal distance of 9.0 m from the wall. We need to find the velocity of the pebbles when they reach Juliet's window with only a horizontal component of velocity.

Step 2: Analyze the Problem Physically

To ensure the pebbles arrive with only a horizontal velocity, we need to consider that the vertical velocity component becomes zero at the window. This occurs due to the influence of gravity.

Step 3: Use Kinematic Equations for Vertical Motion

Using the kinematic equation for vertical motion, we have: \( v_y^2 = u_y^2 + 2g h \), where \( v_y = 0 \) at the window, \( h = 8.0 \) m, and \( g = 9.8 \) m/s². We solve for the initial vertical velocity \( u_y \): \( 0 = u_y^2 + 2(-9.8)(8) \) \( u_y^2 = 2(9.8)(8) \) \( u_y = \sqrt{156.8} = 12.52 \) m/s.

Step 4: Calculate Time of Flight

Using the vertical motion: \( v_y = u_y + a t \), where \( v_y = 0 \), \( u_y = 12.52 \) m/s, and \( a = -9.8 \) m/s², we solve for \( t \): \( 0 = 12.52 - 9.8 t \) \( t = 12.52 / 9.8 = 1.28 \) s.

Step 5: Use Horizontal Motion Equation

With horizontal motion, since there is no acceleration, \( u_x \) is constant and covers the entire horizontal distance: \( x = u_x \cdot t \), where \( x = 9.0 \) m, and \( t = 1.28 \) s. We solve for \( u_x \): \( 9.0 = u_x \cdot 1.28 \) \( u_x = 9.0 / 1.28 = 7.03 \) m/s.

Step 6: Final Velocity at the Window

Since the pebbles have only a horizontal component at the window, the final velocity \( v \) when they hit Juliet's window is \( u_x \), which is calculated as 7.03 m/s.

Key Concepts

Kinematic Equations

In physics, kinematic equations form an essential toolkit for solving problems related to motion. They describe the relationship between the different parameters of motion, such as displacement, velocity, time, and acceleration, without involving forces. For example, when a pebble is thrown or projected, its motion can be split into two parts: horizontal and vertical.

There are typically three main kinematic equations used in such situations:

• The first equation describes velocity changes: \( v = u + at \)
• The second relates displacement with initial and final velocity: \( v^2 = u^2 + 2as \)
• The third equation connects displacement, time, and initial velocity: \( s = ut + \frac{1}{2}at^2 \)

In the context of projectile motion, these equations allow us to calculate various characteristics of the object's path, like its initial velocity, final position, or the time it takes to reach a certain point. For our exercise, using these equations helped in determining the initial vertical speed needed for the pebbles to reach Juliet's window with the desired horizontal velocity.

Horizontal Velocity

Horizontal velocity refers to the speed at which an object moves along the horizontal axis. In projectile motion, it is important to note that the horizontal component of velocity remains constant when air resistance is negligible. This is because there is no horizontal acceleration acting once the object is in motion.

In the problem with Romeo and the pebbles, you need to measure how fast the pebbles are moving horizontally by the time they hit the window. Since there is no acceleration acting on them in the horizontal direction, the formula for horizontal motion simplifies to:

• \( x = u_x \cdot t \)

Here, \( x \) denotes horizontal distance, \( u_x \) is the horizontal velocity, and \( t \) is the time of flight. With these components, we can directly compute the horizontal velocity needed for the pebble to travel the distance to Juliet's window within a specific time.

Vertical Motion

Vertical motion in projectile problems is influenced by gravity, causing objects to accelerate downward at a constant rate. This vertical acceleration, typically denoted as \( g \), is approximately \( 9.8 \) m/s². This component of motion decreases the object's upward velocity until it reaches zero, after which it accelerates downward.

In the case presented, the vertical velocity of the pebbles must be precisely managed so that they reach the level of Juliet's window without continuing upward. This constraint makes the vertical velocity zero at the desired height, resulting in the pebbles having only the horizontal component of velocity remaining. We use the kinematic equation:

• \( v_y^2 = u_y^2 + 2g h \)

Here, \( v_y \) is the final vertical velocity (aimed to be zero at the window), \( u_y \) is the initial vertical velocity, and \( h \) is the vertical displacement. By solving for \( u_y \), we ensure that the vertical component behaves appropriately to achieve the exercise's conditions.