/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Soccer Ball A soccer ball is Pro... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 18

Soccer Ball A soccer ball is Problem 17 . kicked from the ground with an initial speed of \(19.5 \mathrm{~m} / \mathrm{s}\) at an upward angle of \(45^{\circ} .\) A player \(55 \mathrm{~m}\) away in the direction of the kick starts running to meet the ball at that instant. What must be his average speed if he is to meet the ball just before it hits the ground? Neglect air resistance. 19\. Stairway A ball rolls horizontally off the top of a stairway with a speed of \(1.52 \mathrm{~m} / \mathrm{s}\). The steps are \(20.3 \mathrm{~cm}\) high and \(20.3 \mathrm{~cm}\) wide. Which step does the ball hit first?

Short Answer

Expert verified
The player's average speed should be 3 m/s.

Step by step solution

01

Analyze the horizontal motion of the ball

The soccer ball is kicked with an initial speed of 19.5 m/s at an angle of 45掳. First, decompose the initial speed into horizontal and vertical components. The horizontal component of the velocity is given by: i\(v_x = v \, \text{cos}(45掳)\)
02

Analyze the vertical motion of the ball

The vertical component of the velocity is given by: \(v_y = v \, \text{sin}(45掳)\)
03

Calculate the time of flight

The time the ball spends in the air is determined by its vertical motion. Use the following equation for the vertical motion under constant acceleration due to gravity (\(g = 9.8 m/s虏ewline \)):\[\Delta y = v_y t - \frac{1}{2}gt^2\] where the total displacement in the vertical direction (\(\Delta y\)) is zero (the ball lands at the same height it was kicked). Solve for the time of flight (t): \[ 0 = v \, \text{sin}(45掳)t - \frac{1}{2}gt^2\]
04

Solve for time of flight

Re-arrange and solve the time of flight equation: \[ t = \frac{2 v \, \text{sin}(45掳)}{g} \]
05

Calculate Horizontal Distance Covered

Having the time of flight \(t\), compute the horizontal distance covered by the soccer ball: \(d = v_x t\)
06

Determine the Player's Required Speed

The player needs to cover a distance of 55 meters within the same time duration that the ball is in the air. Determine the required speed of the player using: \( v_\text{player} = \frac{55 m}{t}\)
07

Calculate the player's speed

Substitute the value of the time (t) from the earlier computation: (v_player = \(\frac{55 m}{\frac{2\cdot 19.5 m/s \, . \, \text{sin}(45掳)\ \,/ g)}\)\

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Key Concepts

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

Horizontal Motion
Understanding horizontal motion is crucial in projectile motion problems. When dealing with horizontal motion, it鈥檚 important to know that the horizontal velocity of an object remains constant if air resistance is neglected. This is because, in projectile motion, there is no acceleration acting in the horizontal direction after the initial launch. For example, when the soccer ball is kicked, its horizontal velocity can be calculated using the equation:
\( \),
where
\(v = 19.5 \) m/s (the initial speed) and the angle is \( 45掳\).
After breaking down the velocities, we focus on the constant horizontal motion:
  • There is no change in the horizontal speed
  • Horizontal distance covered
    can be found using:
    \(d = v_x \, t \).
    This formula assumes the horizontal speed remains steady throughout the flight time. This is possible in ideal conditions where air resistance is negligible.

The horizontal motion is simple to track as it maintains a constant pace. It only depends on the initial speed and the angle of projection.
Vertical Motion
Vertical motion in projectile problems is a bit more complex because it鈥檚 influenced by the acceleration due to gravity (\(g = 9.8 \, m/s^2 \)). Initial vertical velocity is calculated using:
\( v_y = v \, \text{sin}(45掳) \),
where
\( v = 19.5 \) m/s\br> With this, we can compute how the vertical component of the velocity varies over time:
  • The motion is influenced by gravity, causing the object to decelerate as it rises and accelerate as it falls.
  • The distance or height the object travels vertically can be modeled by the equation:
    \(\text{螖}y = v_y \, t - \, \frac{1}{2} g t^2 \).
    For a ball that returns to the same height level,
    \( \text{螖} y \) will be zero.
  • Solve the quadratic formula for time
    • Rearrange the equation to:
        \( \text{0 = v_y \, t - \, \frac{1}{2} g t^2} \)

      Solve for flight duration
      This equation indicates how vertical motion creates a parabola, describing the ball's rise and fall perfectly.
Time of Flight
The time of flight for a projectile is how long it remains in the air. This time is primarily determined by the vertical component of the motion. Given the initial vertical speed \(v_y \), and the influence of gravity, we use the following steps:
  • Set up the vertical motion equation with
    \( \text{螖}y = 0 \):
    • \(0 = v_y \, t - \, \frac{1}{2} g t^2 \),

This represents the total travel time until it lands.
  • Rearrange to find t:
    \( t = \frac{2 v \, \text{sin}(45掳)}{g} \).
    Here, knowing the initial speed \(19.5 \, m/s \) and
    solving
    will give the total time in air. Once this value is found:
    you can plug it back into the horizontal motion equation to find where the ball lands in the horizontal direction.

    • For example, in the initial problem, knowing this time allows us to then compute how the player must run to intercept the soccer ball just before it lands.

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    Most popular questions from this chapter

    . Stairway A ball rolls horizontally off the top of a stairway with a speed of \(1.52 \mathrm{~m} / \mathrm{s}\). The steps are \(20.3 \mathrm{~cm}\) high and \(20.3 \mathrm{~cm}\) wide. Which step does the ball hit first?

    Watermelon Seed A watermelon seed has the following coordinates: \(x=-5.0 \mathrm{~m}\) and \(y=8.0 \mathrm{~m} .\) Find its position vector (a) in unit- vector notation and as (b) a magnitude and (c) an angle relative to the positive direction of the \(x\) axis. (d) Sketch the vector on a coordinate system. If the seed is moved to the coordinates \((3.00 \mathrm{~m}\), \(0 \mathrm{~m}\) ), what is its displacement (e) in unit-vector notation and as (f) a magnitude and \((\mathrm{g})\) an angle relative to the positive direction of the \(x\) axis?

    \(A\) rifle is aimed horizontally at a target \(30 \mathrm{~m}\) away. The bullet hits the target \(1.9 \mathrm{~cm}\) below the aiming point. What are (a) the bullet's time of flight and (b) its speed as it emerges from the rifle?

    An Earth Satellite An Earth satellite moves in a circular orbit \(640 \mathrm{~km}\) above Earth's surface with a period of \(98.0 \mathrm{~min}\). What are (a) the speed and (b) the magnitude of the centripetal acceleration of the satellite?

    A boy whirls a stone in a horizontal circle of radius \(1.5 \mathrm{~m}\) and at height \(2.0 \mathrm{~m}\) above level ground. The string breaks, and the stone flies off horizontally and strikes the ground after traveling a horizontal distance of \(10 \mathrm{~m}\). What is the magnitude of the centripetal acceleration of the stone while in circular motion?
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