An archer standing on a 15° slope shoots an arrow 20° above the horizontal, as shown in FIGURE CP4.82. How far down the slope does the arrow hit if it is shot with a speed of 5.0 m/s from 1.75 m above the ground?

Name: Physics for Scientists and Engineers
Author: Knight Calc
ISBN: 9780137344796

Verified step by step guidance

Step 1: Break the initial velocity into components. The arrow is shot at an angle of 20° above the horizontal with a speed of 5.0 m/s. Use trigonometric functions to find the horizontal and vertical components of the velocity: $ v_x = v \cdot \cos(20°) $ and $ v_y = v \cdot \sin(20°) $.

Step 2: Set up the equations of motion. The motion of the arrow can be described using the kinematic equations. For the vertical motion, use $ y = y_0 + v_y \cdot t - \frac{1}{2} g \cdot t^2 $, where $ y_0 = 1.75 \, \text{m} $ is the initial height, $ g = 9.8 \, \text{m/s}^2 $ is the acceleration due to gravity, and $ t $ is the time. For the horizontal motion, use $ x = v_x \cdot t $.

Step 3: Account for the slope. The slope is inclined at 15° below the horizontal. The equation of the slope can be expressed as $ y = -x \cdot \tan(15°) $. Combine this with the equations of motion to find the point where the arrow intersects the slope.

Step 4: Solve for the time of flight. Substitute $ y = -x \cdot \tan(15°) $ into the vertical motion equation and use the horizontal motion equation to express $ x $ in terms of $ t $. Solve the resulting equation for $ t $.

Step 5: Calculate the distance down the slope. Once $ t $ is determined, use the horizontal motion equation to find $ x $, and then use the slope equation $ y = -x \cdot \tan(15°) $ to find the corresponding $ y $. The distance down the slope can be calculated using the Pythagorean theorem: $ d = \sqrt{x^2 + y^2} $.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Projectile Motion

Projectile motion refers to the motion of an object that is launched 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 arrow involves calculating its horizontal and vertical displacements over time.

Inclined Plane

An inclined plane is a flat surface that is tilted at an angle to the horizontal. In this scenario, the slope of 15° affects the arrow's landing position. The angle of the slope must be considered when calculating the distance the arrow travels down the slope, as it alters the effective gravitational force acting on the projectile and influences its trajectory.

Initial Velocity Components

The initial velocity of the arrow can be broken down into its horizontal and vertical components using trigonometric functions. Given the launch angle of 20° above the horizontal, the horizontal component is found using cosine, while the vertical component is found using sine. These components are crucial for determining the time of flight and the distance traveled in both the horizontal and vertical directions.

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