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 50c

Chapter 4, Problem 50c

A ball is thrown toward a cliff of height h with a speed of 30 m/s and an angle of 60° above horizontal. It lands on the edge of the cliff 4.0 s later. What is the ball's impact speed?

Verified step by step guidance

Step 1: Break the initial velocity into horizontal and vertical components. Use the equations: \( v_{x0} = v_0 \cos(\theta) \) and \( v_{y0} = v_0 \sin(\theta) \), where \( v_0 = 30 \, \text{m/s} \) and \( \theta = 60^\circ \).

Step 2: Calculate the vertical velocity at the time of impact using the kinematic equation: \( v_y = v_{y0} - g t \), where \( g = 9.8 \, \text{m/s}^2 \) and \( t = 4.0 \, \text{s} \).

Step 3: The horizontal velocity remains constant throughout the motion because there is no horizontal acceleration. Thus, \( v_x = v_{x0} \).

Step 4: Use the Pythagorean theorem to find the magnitude of the impact speed. The total speed is given by \( v = \sqrt{v_x^2 + v_y^2} \).

Step 5: Combine the results from Steps 2 and 3 into the equation from Step 4 to determine the ball's impact speed.

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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. The horizontal motion is uniform, while the vertical motion is influenced by gravity, leading to a parabolic trajectory. Understanding these components is essential for calculating the object's position and speed at any point in time.

Kinematic Equations

Kinematic equations describe the relationships between an object's displacement, velocity, acceleration, and time. For projectile motion, these equations can be applied separately to the horizontal and vertical components. They allow us to calculate final velocities and positions based on initial conditions and the effects of gravity, which is crucial for determining the impact speed of the ball in this scenario.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In the context of projectile motion, the velocity of the ball just before impact can be found by adding its horizontal and vertical velocity components. Since these components are perpendicular to each other, the Pythagorean theorem is used to find the magnitude of the resultant velocity, which represents the ball's impact speed.

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Textbook Question

You are watching an archery tournament when you start wondering how fast an arrow is shot from the bow. Remembering your physics, you ask one of the archers to shoot an arrow parallel to the ground. You find the arrow stuck in the ground 60 m away, making a 30° angle with the ground. How fast was the arrow shot?

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Textbook Question

A ball is thrown toward a cliff of height h with a speed of 30 m/s and an angle of 60° above horizontal. It lands on the edge of the cliff 4.0 s later. What was the maximum height of the ball?

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Textbook Question

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