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Ch 03: Motion in Two or Three Dimensions
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 03: Motion in Two or Three DimensionsProblem 21d
Chapter 3, Problem 21d

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 Draw x-t, y-t, vx–t, and vy–t graphs for the motion.

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Step 1: Begin by analyzing the problem. The rock is thrown from a height of 15.0 meters with an initial speed of 30.0 m/s at an angle of 33.0° above the horizontal. We need to consider the motion in both the horizontal (x) and vertical (y) directions separately.
Step 2: Calculate the initial velocity components. Use trigonometry to find the horizontal and vertical components of the initial velocity. The horizontal component (υx) is given by υx = υ * cos(θ), and the vertical component (υy) is given by υy = υ * sin(θ), where υ is the initial speed and θ is the angle of projection.
Step 3: Determine the equations of motion. For horizontal motion, the velocity remains constant since there is no air resistance, so x(t) = υx * t. For vertical motion, use the kinematic equation y(t) = y0 + υy * t - (1/2) * g * t^2, where y0 is the initial height and g is the acceleration due to gravity (approximately 9.81 m/s²).
Step 4: Plot the x-t and y-t graphs. The x-t graph will be a straight line indicating constant velocity, while the y-t graph will be a parabola indicating the effect of gravity on the vertical motion. The initial height and the time of flight will influence the shape of the y-t graph.
Step 5: Plot the υx-t and υy-t graphs. The υx-t graph will be a horizontal line since the horizontal velocity is constant. The υy-t graph will be a linear graph starting at the initial vertical velocity and decreasing linearly due to gravity, eventually becoming negative as the rock falls back down.

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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 thrown or projected into the air, subject to only the acceleration of gravity. It involves two components: horizontal and vertical motion, which are independent of each other. Understanding the initial velocity, angle of projection, and height is crucial for analyzing the trajectory and determining the object's position and velocity at any point in time.
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Kinematic Equations

Kinematic equations describe the motion of objects in terms of displacement, velocity, acceleration, and time. For projectile motion, these equations help calculate the horizontal and vertical components of motion separately. They are essential for determining the position and velocity of the rock at any given time, which is necessary for plotting the x-t, y-t, υx–t, and υy–t graphs.
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Graphical Analysis of Motion

Graphical analysis involves plotting graphs to visually represent the motion of an object. The x-t and y-t graphs show how the horizontal and vertical positions change over time, while the υx–t and υy–t graphs depict the changes in horizontal and vertical velocities. These graphs provide insights into the motion's characteristics, such as constant horizontal velocity and changing vertical velocity due to gravity.
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Related Practice
Textbook Question

In a carnival booth, you can win a stuffed giraffe if you toss a quarter into a small dish. The dish is on a shelf above the point where the quarter leaves your hand and is a horizontal distance of 2.1 m from this point (Fig. E3.19). If you toss the coin with a velocity of 6.4 m/s at an angle of 60° above the horizontal, the coin will land in the dish. Ignore air resistance. What is the height of the shelf above the point where the quarter leaves your hand?

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Textbook Question

The earth has a radius of 6380 km and turns around once on its axis in 24 h. If arad at the equator is greater than g, objects will fly off the earth's surface and into space. (We will see the reason for this in Chapter 5.) What would the period of the earth's rotation have to be for this to occur?

Textbook Question

In a carnival booth, you can win a stuffed giraffe if you toss a quarter into a small dish. The dish is on a shelf above the point where the quarter leaves your hand and is a horizontal distance of 2.1 m from this point (Fig. E3.19). If you toss the coin with a velocity of 6.4 m/s at an angle of 60° above the horizontal, the coin will land in the dish. Ignore air resistance. What is the vertical component of the velocity of the quarter just before it lands in the dish?

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Textbook Question

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The earth has a radius of 6380 km and turns around once on its axis in 24 h. What is the radial acceleration of an object at the earth's equator? Give your answer in m/s2 and as a fraction of g.

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