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 50b

Chapter 4, Problem 50b

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?

Verified step by step guidance

Step 1: Break the initial velocity into its horizontal and vertical components using trigonometric functions. The vertical component of velocity is given by \( v_{y0} = v_0 \sin \theta \), and the horizontal component is \( v_{x0} = v_0 \cos \theta \). Here, \( v_0 = 30 \, \text{m/s} \) and \( \theta = 60^\circ \).

Step 2: Use the kinematic equation for vertical motion to find the time at which the ball reaches its maximum height. At maximum height, the vertical velocity becomes zero, so \( v_y = v_{y0} - g t \), where \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)). Solve for \( t \) when \( v_y = 0 \).

Step 3: Calculate the maximum height using the kinematic equation \( y = v_{y0} t - \frac{1}{2} g t^2 \). Substitute the time \( t \) found in Step 2 and the vertical velocity \( v_{y0} \) into this equation.

Step 4: Add the initial height of the ball (if any) to the calculated height from Step 3. In this case, the ball starts from ground level, so the initial height is zero.

Step 5: Verify the result conceptually by ensuring the calculated maximum height is consistent with the given time of flight and the motion of the ball. The total time of flight (4.0 s) should include the time to reach maximum height and the time to descend from it.

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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 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 essential for solving problems related to projectiles.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. For projectile motion, these equations can be used to relate displacement, initial velocity, final velocity, acceleration, and time. In this context, they help calculate the maximum height reached by the ball by analyzing its vertical motion separately from its horizontal motion.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are crucial for resolving the initial velocity of the projectile into its horizontal and vertical components. For an angle of 60°, the vertical component can be found using the sine function, while the horizontal component uses the cosine function. These components are essential for applying kinematic equations to determine the ball's trajectory and maximum height.

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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 projectile is launched from ground level at angle θ and speed v₀ into a headwind that causes a constant horizontal acceleration of magnitude a opposite the direction of motion. Find an expression in terms of a and g for the launch angle that gives maximum range.

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Textbook Question

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. At what angle should you toss the ball?

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Textbook Question

A gray kangaroo can bound across level ground with each jump carrying it 10 m from the takeoff point. Typically the kangaroo leaves the ground at a 20° angle. If this is so: What is its maximum height above the ground?

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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 is the ball's impact speed?

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Textbook Question

A projectile is launched from ground level at angle θ and speed v₀ into a headwind that causes a constant horizontal acceleration of magnitude a opposite the direction of motion. What is the angle for maximum range if a is 10% of g?