Ch 04: Kinematics in Two Dimensions

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 04: Kinematics in Two Dimensions

Problem 59c

Chapter 4, Problem 59c

A cannonball is fired at 100 m/s from a barrel tilted upward at 25°. What is the angle after the cannonball travels 500 m?

Verified step by step guidance

Step 1: Recognize that the problem involves projectile motion. The angle after the cannonball travels 500 m refers to the angle of the velocity vector relative to the horizontal at that point in its trajectory.

Step 2: Break the initial velocity into horizontal and vertical components using trigonometric functions. The horizontal velocity component is \( v_x = v \cdot \cos(\theta) \), and the vertical velocity component is \( v_y = v \cdot \sin(\theta) \), where \( v = 100 \, \text{m/s} \) and \( \theta = 25^\circ \).

Step 3: Use the kinematic equation for vertical motion to find the vertical velocity \( v_y \) at the point where the cannonball has traveled 500 m horizontally. The equation is \( v_y = v_{y0} - g \cdot t \), where \( v_{y0} \) is the initial vertical velocity, \( g \) is the acceleration due to gravity (\( 9.8 \; \text{m/s}^2 \)), and \( t \) is the time of flight. To find \( t \), use the horizontal motion equation \( x = v_x \cdot t \), where \( x = 500 \; \text{m} \).

Step 4: Once \( v_x \) and \( v_y \) are determined, calculate the angle \( \phi \) of the velocity vector relative to the horizontal using the formula \( \tan(\phi) = \frac{v_y}{v_x} \). Solve for \( \phi \) using \( \phi = \arctan\left(\frac{v_y}{v_x}\right) \).

Step 5: Interpret the result. The angle \( \phi \) represents the direction of the cannonball's velocity vector relative to the horizontal after it has traveled 500 m horizontally. This angle will be less than the initial firing angle of 25° due to the effects of gravity on the vertical component of velocity.

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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 gravitational forces. It can be analyzed by breaking it into horizontal and vertical components, allowing us to predict the object's trajectory, range, and time of flight. Understanding the principles of projectile motion is essential for solving problems involving objects like cannonballs.

Initial Velocity Components

The initial velocity of a projectile can be divided into horizontal and vertical components using trigonometric functions. For a cannonball fired at an angle, the horizontal component is found using cosine, while the vertical component is found using sine. These components are crucial for determining the projectile's path and position at any given time.

Angle of Projection

The angle of projection is the angle at which an object is launched relative to the horizontal. It significantly affects the range and height of the projectile. In this problem, understanding how the angle changes as the cannonball travels is key to determining its position after covering a specific distance, such as 500 m.

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