A man stands on the roof of a 15.0-m-tall building and throws a rock with a speed of 30.0 m/s at an angle of 33.0° above the horizontal. Ignore air resistance. Calculate (d) Draw x-t, y-t, υx–t, and υy–t graphs for the motion.

Verified step by step guidance

1. Identify the initial conditions for the projectile motion. The initial velocity (v_0) is 30.0 m/s and the angle of projection (\theta) is 33.0°. The initial height (y_0) from which the rock is thrown is 15.0 m.
2. Decompose the initial velocity into horizontal (v_{0x}) and vertical (v_{0y}) components. Use the equations v_{0x} = v_0 \cos(\theta) and v_{0y} = v_0 \sin(\theta).
3. For the x-t graph (horizontal position vs. time), use the equation x(t) = v_{0x} \cdot t, since there is no horizontal acceleration.
4. For the y-t graph (vertical position vs. time), use the equation y(t) = y_0 + v_{0y} \cdot t - \frac{1}{2} g t^2, where g is the acceleration due to gravity (approximately 9.81 m/s^2).
5. For the \upsilon_x-t graph (horizontal velocity vs. time), note that horizontal velocity remains constant (\upsilon_x = v_{0x}) as there is no horizontal acceleration. For the \upsilon_y-t graph (vertical velocity vs. time), use the equation \upsilon_y(t) = v_{0y} - g \cdot t.

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 components of initial velocity and the effects of gravity is crucial for solving projectile motion problems.

Kinematic Equations

Kinematic equations describe the relationship between an object's displacement, velocity, acceleration, and time. These equations are essential for analyzing motion in both the x (horizontal) and y (vertical) directions. In projectile motion, separate kinematic equations are applied to each direction, allowing for the calculation of various parameters such as time of flight, maximum height, and range.

Graphical Representation of Motion

Graphical representation of motion involves plotting graphs to visualize the relationships between different physical quantities over time. In this context, x-t (position vs. time), y-t (position vs. time), υx-t (horizontal velocity vs. time), and υy-t (vertical velocity vs. time) graphs help illustrate how the position and velocity of the projectile change throughout its flight. These graphs provide insights into the motion's characteristics, such as symmetry and the effects of gravity.