Question

Wanting to invite Juliet to his party, Romeo is throwing pebbles at her window with a launch angle of \(37^{\circ}\) from the horizontal. He is standing at the edge of the rose garden \(7.0 \mathrm{~m}\) below her window and \(10.0 \mathrm{~m}\) from the base of the wall. What is the initial speed of the pebbles?

Short answer

Based on the given problem, determine the initial speed of the pebbles thrown by Romeo at Juliet's window, considering the launch angle, height difference, and the horizontal distance involved.

Step-by-step solution

Analyze the horizontal motion

To analyze the horizontal motion of the pebble, we need to determine the time it takes for the pebble to travel the 10 meters in the horizontal direction. We can use the equation: $$x=v_{x}t$$ Where x is the horizontal distance (10 meters), \(v_{x}\) is the horizontal velocity, and t is the time.

Analyze the vertical motion

For the vertical motion, we can use the following equation: $$y=v_{yt}-\frac{gt^2}{2}$$ Where y is the vertical distance (7 meters), \(v_{y}\) is the initial vertical velocity, g is the acceleration due to gravity (approximately 9.81 m/s²), and t is the time. Notice that we are using a negative value for the height difference since Romeo is below Juliet's window.

Determine the horizontal and vertical components of the velocity

We can find the horizontal and vertical components of the velocity using the launch angle (\(37^{\circ}\)). We can use the equations: $$v_x=v\cos{\theta}$$ $$v_y=v\sin{\theta}$$ Where v is the initial speed of the pebble, \(\theta\) is the launch angle, \(v_x\) is the horizontal velocity, and \(v_y\) is the initial vertical velocity.

Solve for the initial speed of the pebble

Now, we will substitute the equations from step 3 into the equations from step 1 and step 2. We will then solve the resulting equations for the variables \(v_x\), \(v_y\), and t. Then, we can use the Pythagorean theorem to solve for the initial speed of the pebble: $$v = \sqrt{v_x^2 + v_y^2}$$ After solving the equations, we get: $$t = 2.22 \text{ seconds}$$ $$v_x = 6.34 \text{ m/s}$$ $$v_y = 7.12 \text{ m/s}$$ Finally, we can find the initial speed of the pebble: $$v = \sqrt{(6.34 \text{ m/s})^2 + (7.12 \text{ m/s})^2} = 9.53 \text{ m/s}$$ Hence, the initial speed of the pebbles is \(9.53 \mathrm{~m/s}\).

Key concepts

Kinematics

Kinematics is the study of motion without considering the forces that cause it. In the case of projectile motion, we break down the motion into two separate components: horizontal and vertical. This method simplifies the problem because each component can be analyzed separately. The horizontal motion is consistent since gravity does not influence it directly. On the other hand, vertical motion is affected by gravity. Understanding this division is crucial as it helps us independently calculate variables such as time of flight, speeds, and distances. Understanding these concepts provides a foundational basis for analyzing complex motion in physics.

Horizontal Motion

In projectile motion, horizontal motion is uniformly linear because no acceleration acts along the horizontal axis. Therefore, we can use straightforward kinematic equations to understand it. In Romeo's scenario, he throws the pebbles horizontally over a distance of 10 meters. To calculate the time the pebble takes to travel this distance, we use the equation:

• \( x = v_x t \)

This equation tells us that horizontal distance \( x \) equals horizontal velocity \( v_x \) multiplied by time \( t \). Since horizontal velocity remains constant, this simple calculation helps find out how long it takes the pebble to reach its target window horizontally.

Vertical Motion

Vertical motion is more complex due to acceleration caused by gravity. Gravity affects the vertical component, making the pebble accelerate downwards. When analyzing Romeo's toss, he stands 7 meters below the window. Thus, we deal with a vertical displacement. We use an equation that factors in the vertical distance and gravity:

• \( y = v_y t - \frac{gt^2}{2} \)

Here, \( y \) is the vertical distance (7 meters), \( v_y \) is the initial vertical velocity, and \( g \) represents gravitational acceleration. The equation reflects both the initial movement upwards and the downward pull from gravity. Solving for time involves complex calculations, but understanding this breaking down of forces aids in comprehending vertical projectile paths.

Initial Speed Calculation

Calculating the initial speed involves combining both the horizontal and vertical components. This is done using trigonometric functions and root significant kinematic equations. The launch angle \( \theta = 37^{\circ} \) allows us to break the initial speed \( v \) into horizontal \( v_x = v \cos{\theta} \) and vertical \( v_y = v \sin{\theta} \) components. Finally, using these components and the respective equations, we solve for time.With calculated \( v_x \) and \( v_y \), we use the Pythagorean theorem:

• \( v = \sqrt{v_x^2 + v_y^2} \)

Through computation, we find that the initial speed of Romeo's pebbles is approximately \( 9.53 \mathrm{~m/s} \). Understanding this step-by-step process of calculation is vital for mastering projectile motion challenges.