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Chapter 4: Problem 31

In a jump spike, a volleyball player slams the ball from overhead and toward the opposite floor. Controlling the angle of the spike is difficult. Suppose a ball is spiked from a height of 2.30 \(\mathrm{m}\) with an initial speed of \(20.0 \mathrm{~m} / \mathrm{s}\) at a downward angle of \(18.00^{\circ} .\) How much farther on the opposite floor would it have landed if the downward angle were, instead, \(8.00^{\circ} ?\)

Short Answer

Expert verified
With a downward angle of 8.00掳, the ball lands 1.73 m farther than at 18.00掳.

Step by step solution

01

Analyze the Problem

The problem involves the projectile motion of a volleyball being spiked at two different angles but with the same initial speed. Our goal is to calculate how much farther the ball lands with a smaller downward angle of 8.00掳 compared to 18.00掳.
02

Define the Equations of Motion

To solve the problem, we'll use kinematic equations for projectile motion. We'll need to separate the initial velocity into horizontal and vertical components. These are calculated as follows:- Horizontal velocity: \( v_{x} = v_{0} \cos(\theta) \)- Vertical velocity: \( v_{y} = v_{0} \sin(\theta) \).
03

Calculate the Time of Flight for Each Angle

To find the time of flight, use the vertical motion equation: \( y - y_0 = v_{y} t + \frac{1}{2} g t^2 \), where \( g = -9.8 \, \text{m/s}^2 \), \( y = 0 \, \text{m} \), and \( y_0 = 2.30 \, \text{m} \). Solve the quadratic equation for \( t \) using the vertical velocity components for each angle.
04

Find Horizontal Range for 18.00掳

Substitute the time of flight for the 18.00掳 angle \( t_{18} \) into the equation for horizontal motion: \( x = v_{x} \cdot t \), where \( v_{x} = 20.0 \cos(18.00掳) \). Calculate the horizontal distance covered.
05

Find Horizontal Range for 8.00掳

Substitute the time of flight for the 8.00掳 angle \( t_{8} \) into the horizontal motion equation. Recalculate the horizontal distance with \( v_{x} = 20.0 \cos(8.00掳) \).
06

Calculate Difference in Ranges

Subtract the horizontal range of the 18.00掳 spike from the horizontal range of the 8.00掳 spike to determine how much farther the ball lands with the smaller angle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Kinematic Equations
Kinematic equations are essential in analyzing projectile motion, especially in sports science. They allow us to predict and describe how objects move under the influence of gravity. In projectile motion problems like the volleyball spike, we use them to understand the ball's trajectory, speed, and the way it moves in both horizontal and vertical directions.
  • They provide a mathematical framework to describe the object's movement.
  • They can solve for unknowns such as time, distance, and velocity.
The key to applying these equations lies in breaking down the motion into two parts鈥攈orizontal and vertical. Each part is analyzed independently, which makes it easier to predict the object's complete path.
Understanding how to categorize these components is crucial for solving projectile motion problems effectively, as it simplifies the calculations.
Calculating Horizontal Range
The horizontal range in projectile motion represents how far an object travels along the horizontal plane before it hits the ground. In the volleyball example, it's the distance the ball covers from the point of impact to where it touches the ground. Calculating this range involves knowing the horizontal component of the initial velocity and the time of flight.
Several factors affect the horizontal range:
  • The initial horizontal velocity, which depends on the angle at which the ball is spiked.
  • The time the ball stays airborne. Longer time in motion increases the horizontal range.
  • The absence of horizontal acceleration, simplifying horizontal range calculations to a constant velocity multiplied by time.
For the volleyball spike, changes in the angle alter these parameters, resulting in different landing distances on the opposite floor. By calculating the range for each angle, we can understand how the projectile's trajectory changes.
Breaking Down Initial Velocity Components
The initial velocity of a projectile can be divided into two perpendicular components: horizontal and vertical. These components determine how the projectile will move in each direction. In the case of the spiked volleyball:
  • The horizontal component, denoted as \(v_x = v_0 \cos(\theta)\), indicates how far the ball will travel across the floor.
  • The vertical component, denoted as \(v_y = v_0 \sin(\theta)\), determines how the ball ascends and descends under gravity's influence.
Understanding these components helps in solving various parts of the problem. For instance, calculating time of flight requires focusing on the vertical component to see how long the object stays in the air.
The angle of launch critically affects these components; a steeper angle yields a greater vertical component, while a shallower angle results in a stronger horizontal component, affecting the overall range and behavior of the motion.

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Most popular questions from this chapter

A woman who can row a boat at \(6.4 \mathrm{~km} / \mathrm{h}\) in still water faces a long, straight river with a width of \(6.4 \mathrm{~km}\) and a current of \(3.2 \mathrm{~km} / \mathrm{h}\). Let i point directly across the river and j point directly downstream. If she rows in a straight line to a point directly opposite her starting position, (a) at what angle to 矛 must she point the boat and (b) how long will she take? (c) How long will she take if, instead, she rows \(3.2 \mathrm{~km}\) down the river and then back to her starting point? (d) How long if she rows \(3.2 \mathrm{~km}\) up the river and then back to her starting point? (e) At what angle to i should she point the boat if she wants to cross the river in the shortest possible time? (f) How long is that shortest time?

A plane flies \(483 \mathrm{~km}\) east from city \(A\) to city \(B\) in \(45.0 \mathrm{~min}\) and then \(966 \mathrm{~km}\) south from city \(B\) to city \(C\) in \(1.50 \mathrm{~h}\). For the total trip, what are the (a) magnitude and (b) direction of the plane's displacement, the (c) magnitude and (d) direction of its average velocity, and (e) its average speed?

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