You're 6.0 m from one wall of the house seen in FIGURE P4.55. You want to toss a ball to your friend who is 6.0 m from the opposite wall. The throw and catch each occur 1.0 m above the ground. c. At what angle should you toss the ball?

Verified step by step guidance

1. Identify the total horizontal distance the ball needs to travel. Since you are 6.0 m from one wall and your friend is 6.0 m from the opposite wall, add these distances to the width of the house (if given, or assume a value if not specified).
2. Use the projectile motion equation to relate the horizontal distance (range), initial velocity, and angle of projection. The equation is: $R=\frac{v^2\sin (2\theta )}{g}$, where $R$ is the range, $v$ is the initial velocity, $\theta$ is the angle of projection, and $g$ is the acceleration due to gravity (approximately 9.8 m/s^2).
3. Since the throw and catch each occur 1.0 m above the ground, you can assume that the vertical displacement is zero. This simplifies the calculations as the time to reach maximum height equals the time to fall back down.
4. Rearrange the projectile motion equation to solve for the angle $\theta$. The equation becomes: $\theta =\frac{1}{2}\arcsin \left(\frac{gR}{v^2}\right)$.
5. Choose an appropriate initial velocity based on typical throwing speeds or through trial and error, and calculate the angle using the rearranged equation.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is thrown into the air and is subject to the force of gravity. It can be analyzed in two dimensions: horizontal and vertical. The horizontal motion is uniform, while the vertical motion is influenced by gravitational acceleration. Understanding the trajectory of the ball involves calculating the initial velocity, angle of projection, and the time of flight.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In the context of projectile motion, these equations can be used to relate the initial velocity, angle of projection, time of flight, and displacement. For this problem, the horizontal and vertical components of the motion can be analyzed separately to determine the angle needed for the ball to reach the desired height and distance.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are essential for analyzing angles and distances in projectile motion. The angle of projection affects both the horizontal and vertical components of the initial velocity. By using these functions, one can resolve the initial velocity into its components and apply them to the kinematic equations to find the optimal angle for the toss.