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Ch 03: Motion in Two or Three Dimensions
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 03: Motion in Two or Three DimensionsProblema 19b
Capitolo 3, Problema 19b

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?
A person tossing a quarter at 60° to hit a dish 2.1 m away, aiming to win a stuffed giraffe.

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First, resolve the initial velocity of the quarter into its horizontal and vertical components. The horizontal component \( v_{x} \) can be found using \( v_{x} = v \cdot \cos(\theta) \), where \( v = 6.4 \text{ m/s} \) and \( \theta = 60^{\circ} \). The vertical component \( v_{y} \) is given by \( v_{y} = v \cdot \sin(\theta) \).
Next, calculate the time \( t \) it takes for the quarter to travel the horizontal distance to the dish. Use the formula \( t = \frac{d}{v_{x}} \), where \( d = 2.1 \text{ m} \) is the horizontal distance.
Now, determine the vertical component of the velocity just before the quarter lands in the dish. Use the kinematic equation \( v_{y_{f}} = v_{y} - g \cdot t \), where \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity.
Consider the direction of the vertical component of the velocity. Since the quarter is moving upwards initially and then downwards, the final vertical velocity will be directed downwards.
Finally, ensure that the calculated vertical component of the velocity is consistent with the direction of motion just before landing in the dish, which should be downward.

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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 gravity, resulting in a parabolic trajectory. Understanding this concept is crucial for determining the path and final velocity of the quarter as it travels to the dish.
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Introduction to Projectile Motion

Components of Velocity

Velocity can be broken down into horizontal and vertical components, especially when dealing with angled launches. For a projectile launched at an angle, the initial velocity can be split into two parts: the horizontal component (v_x) and the vertical component (v_y). These components can be calculated using trigonometric functions, where v_x = v * cos(θ) and v_y = v * sin(θ). This breakdown is essential for analyzing the quarter's motion in both dimensions.
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Calculating Velocity Components

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration, such as gravity. These equations relate displacement, initial velocity, final velocity, acceleration, and time. In this scenario, the vertical motion of the quarter can be analyzed using these equations to find the vertical component of the velocity just before it lands in the dish. The relevant equation for vertical motion is v_y^2 = v_{y0}^2 + 2a_y d_y, where a_y is the acceleration due to gravity.
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