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Chapter 2: Problem 229

In a test of vertical leaping ability, a basketball player (a) crouches just before jumping, (b) has given his mass center \(G\) a vertical velocity \(v_{0}\) at the instant his feet leave the surface, and \((c)\) reaches the maximum height. If the player can raise his mass center 3 feet as shown, estimate the initial velocity \(v_{0}\) of his mass center in position ( \(b\) ).

Short Answer

Expert verified
The initial velocity of the player's mass center is approximately 13.9 ft/s.

Step by step solution

01

Understand the Problem

The basketball player jumps from a crouched position, reaching a maximum vertical height of 3 feet above the original position. We need to determine the initial velocity \(v_0\) of the basketball player's mass center when his feet leave the ground.
02

Recall the Kinematic Equation

We can use the kinematic equation for motion under constant acceleration (gravity in this case) to relate the initial velocity, final velocity, and displacement. This equation is: \( v^2 = v_0^2 + 2a s \), where \( v \) is the final velocity, \( v_0 \) is the initial velocity, \( a \) is the acceleration, and \( s \) is the displacement.
03

Substitute Known Values

At the highest point of the jump, the final velocity \( v \) is 0 because momentarily the player stops ascending. The displacement \( s \) is 3 feet. The acceleration \( a \) is \(-32.2 \text{ ft/s}^2\) (negative due to gravity). Thus, we substitute these values into the equation: \( 0 = v_0^2 + 2(-32.2)(3) \).
04

Solve for Initial Velocity \(v_0\)

Rearrange the equation to solve for \(v_0^2\): \( v_0^2 = 2 \times 32.2 \times 3 \). Calculating the right side, we get \( v_0^2 = 193.2 \). Taking the square root gives \( v_0 = \sqrt{193.2} \approx 13.9 \text{ ft/s} \).

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Key Concepts

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

Initial Velocity
The initial velocity is a fundamental concept in kinematics, particularly important in analyzing projectile motion. It represents the speed at which an object starts its motion. In the context of our exercise, it's the speed of the basketball player's mass center right as his feet leave the ground. This velocity is crucial as it influences how high or far the player can jump.

To calculate the initial velocity (\(v_0\)) for a vertical leap, we typically use the kinematic equation:\[v^2 = v_0^2 + 2a s\]Here:
  • \(v_0\) is the initial velocity we want to find.
  • \(v\) is the final velocity, which is 0 at the top of the leap because the player's vertical motion momentarily stops.
  • \(a\) is the acceleration due to gravity (-32.2 ft/s虏).
  • \(s\) is the displacement, which in this exercise is the leap's height (3 feet).
Understanding how to work with initial velocity and kinematic equations allows us to predict and analyze motion effectively. Initial velocity determines how far an object can travel, especially in situations governed by gravity.
Vertical Leap
A vertical leap is a jumping motion where the main objective is to move upwards against the force of gravity. This action showcases an individual's ability to accelerate their mass away from the ground. Athletes often aim to maximize their vertical leap to improve performance in sports like basketball and volleyball.

In the exercise, the basketball player's vertical leap increased his mass center by 3 feet. Understanding how to calculate and improve vertical leap involves grasping foundational physics concepts like initial velocity and acceleration. Vertical leaps are influenced by multiple factors:
  • Muscle strength and explosive power.
  • Technique and positioning during takeoff.
  • Body composition and mass distribution.
  • Proper force application from feet to the ground.
During a jump, the player must overcome static positioning and gravity, using muscles to propel their body upwards. By optimizing the initial velocity, athletes can achieve greater vertical distances, showcasing the importance of both physiology and kinematics in vertical leaps.
Acceleration Due to Gravity
When analyzing motion, especially vertical leaps, understanding the concept of acceleration due to gravity (\(g\)) is key. It's a constant value that represents the rate at which objects accelerate towards the Earth. On Earth's surface, this value is approximately -32.2 feet per second squared. The negative sign indicates that gravity acts in the opposite direction to the jump, pulling the jumper back down.

Gravity plays a critical role in limiting how high someone can leap. It affects:
  • The maximum height attained during a jump.
  • The time spent in the air.
In our calculation, gravity's constant rate is used in the kinematic equation to find the initial velocity. By solving \[v^2 = v_0^2 + 2a s\] with \(a\) as \(-32.2 \text{ ft/s}^2\), we see how gravity influences the motion. All objects, regardless of mass, experience the same acceleration due to gravity when in air. Understanding gravity's impact helps athletes and engineers design strategies and equipment to work efficiently against this natural force.

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Most popular questions from this chapter

The preliminary design for a "small" space station to orbit the earth in a circular path consists of a ring (torus) with a circular cross section as shown. The living space within the torus is shown in section \(A,\) where the "ground level" is \(20 \mathrm{ft}\) from the center of the section. Calculate the angular speed \(N\) in revolutions per minute required to simulate standard gravity at the surface of the earth \(\left(32.17 \mathrm{ft} / \mathrm{sec}^{2}\right) .\) Recall that you would be unaware of a gravitational field if you were in a nonrotating spacecraft in a circular orbit around the earth.

A particle \(P\) is launched from point \(A\) with the initial conditions shown. If the particle is subjected to aerodynamic drag, compute the range \(R\) of the particle and compare this with the case in which aerodynamic drag is neglected. Plot the trajectories of the particle for both cases. Use the values \(v_{0}=\) \(65 \mathrm{m} / \mathrm{s}, \theta=35^{\circ},\) and \(k=4.0 \times 10^{-3} \mathrm{m}^{-1}\). (Note: The acceleration due to aerodynamic drag has the form \(\mathbf{a}_{D}=-k v^{2} \mathbf{t},\) where \(k\) is a positive constant, \(v\) is the particle speed, and \(t\) is the unit vector associated with the instantaneous velocity \(\mathbf{v}\) of the particle. The unit vector \(t\) has the form \(t=\frac{v_{x} \mathbf{i}+v_{y} \mathbf{j}}{\sqrt{v_{x}^{2}+v_{y}^{2}}},\) where \(v_{x}\) and \(v_{y}\) are the instantaneous \(x\) -and \(y\) -components of particle velocity, respectively.)

A vacuum-propelled capsule for a high-speed tube transportation system of the future is being designed for operation between two stations \(A\) and \(B,\) which are \(10 \mathrm{km}\) apart. If the acceleration and deceleration are to have a limiting magnitude of \(0.6 g\) and if velocities are to be limited to \(400 \mathrm{km} / \mathrm{h}\) determine the minimum time \(t\) for the capsule to make the 10 -km trip.

The rod \(O A\) is held at the constant angle \(\beta=30^{\circ}\) while it rotates about the vertical with a constant angular rate \(\dot{\theta}=120\) rev/min. Simultaneously, the sliding ball \(P\) oscillates along the rod with its distance in millimeters from the fixed pivot \(O\) given by \(R=200+50 \sin 2 \pi n t,\) where the frequency \(n\) of oscillation along the rod is a constant 2 cycles per second and where \(t\) is the time in seconds. Calculate the magnitude of the acceleration of \(P\) for an instant when its velocity along the rod from 0 toward \(A\) is a maximum.

At time \(t=0,\) the 1.8 -lb particle \(P\) is given an initial velocity \(v_{0}=1 \mathrm{ft} / \mathrm{sec}\) at the position \(\theta=0\) and subsequently slides along the circular path of radius \(r=1.5 \mathrm{ft}\). Because of the viscous fluid and the effect of gravitational acceleration, the tangential acceleration is \(a_{t}=g \cos \theta-\frac{k}{m} v,\) where the constant \(k=0.2 \mathrm{lb}\) -sec/ft is a drag parameter. Determine and plot both \(\theta\) and \(\dot{\theta}\) as functions of the time \(t\) over the range \(0 \leq t \leq 5\) sec. Determine the maximum values of \(\theta\) and \(\dot{\theta}\) and the corresponding values of \(t .\) Also determine the first time at which \(\theta=90^{\circ}\)
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