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Chapter 6: Problem 88

Suppose that the resistive force of the air on a skydiver can be approximated by \(f=-b v^{2} .\) If the terminal velocity of a 50.0 -kg skydiver is \(60.0 \mathrm{m} / \mathrm{s}\), what is the value of \(b ?\)

Short Answer

Expert verified
The value of \(b\) is 0.1367 \(kg/m\).

Step by step solution

01

Understand the situation at terminal velocity

At terminal velocity, the skydiver is no longer accelerating. Thus, the total force acting on the skydiver must be zero. Two forces are acting on the skydiver: the gravitational force and the resistive force. Therefore, Gravity force = Resistive force, i.e., \(mg = bv^2\).
02

Apply the values to find the resistive constant

Now replace \(m\), \(g\), and \(v\) with actual values given in the problem to solve for \(b\). The gravitational constant, \(g\), is 9.81 \(m/s^2\). So \(b = \frac{mg}{v^2} = \frac{50kg * 9.81 m/s^2}{(60m/s)^2}\)
03

Calculate the value of b

Now calculate the value of \(b\). After solving the above equation, the calculated value of \(b\) is 0.1367 \(kg/m\).

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Key Concepts

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

Resistive Force
The resistive force is an essential concept to understand when discussing objects in motion through a fluid, like air. In our case, this force is acted upon by air Resistance, which slows down a skydiver. This force is mathematically represented as \( f = -b v^2 \). Here,
  • \(f\) is the resistive force.
  • \(b\) is the resistive constant.
  • \(v\) is the velocity of the object.
The negative sign indicates that the force acts in the opposite direction to the velocity. As the velocity increases, the resistive force, acting against the motion, also increases. This eventually leads to a state where the object reaches a terminal velocity where no further acceleration occurs.
Gravitational Force
Gravitational force plays a vital role in the journey of a skydiver. It is the force due to gravity that pulls objects toward the Earth's center. The gravitational force acting on an object of mass \(m\) near the Earth鈥檚 surface is given by the product of mass \(m\) and the gravitational acceleration \(g\), written as \( F_g = mg \). Here,
  • \(F_g\) is the gravitational force.
  • \(m\) is the mass of the skydiver.
  • \(g\) is the acceleration due to gravity, approximately \(9.81 m/s^2\).
In the context of terminal velocity, the gravitational force must equal the resistive force for the skydiver to stop accelerating. This balance is what ultimately brings the skydiver to a steady terminal velocity.
Resistive Constant
The resistive constant, denoted by \(b\), is a crucial parameter in the equation of resistive force, \(f = -b v^2\). It quantifies the extent of resistance a fluid, like air, imposes on an object moving through it. In the given problem, the value of \(b\) was calculated using the equation at terminal velocity, where all acting forces balance out, such that the gravitational force equals the resistive force. Given that \(mg = bv^2\), it turns out that
  • \(b\) can be expressed as \(b = \frac{mg}{v^2}\).
  • Substituting the mass (50 kg), gravitational acceleration (9.81 m/s虏), and terminal velocity (60 m/s), we find \(b = 0.1367\) kg/m.
The resistive constant, thus, depends on these factors and is specific to the conditions of each unique scenario. It provides insights into how changes in velocity or shape may influence the resistance experienced by the skydiver.

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Most popular questions from this chapter

A child of mass 40.0 kg is in a roller coaster car that travels in a loop of radius \(7.00 \mathrm{m}\). At point A the speed of the car is \(10.0 \mathrm{m} / \mathrm{s},\) and at point \(\mathrm{B},\) the speed is \(10.5 \mathrm{m} / \mathrm{s}\) Assume the child is not holding on and does not wear a seat belt. (a) What is the force of the car seat on the child at point A? (b) What is the force of the car seat on the child at point B? (c) What minimum speed is required to keep the child in his seat at point A?

A boater and motor boat are at rest on a lake. Together, they have mass 200.0 kg. If the thrust of the motor is a constant force of \(40.0 \mathrm{N}\) in the direction of motion, and if the resistive force of the water is numerically equivalent to 2 times the speed \(v\) of the boat, set up and solve the differential equation to find: (a) the velocity of the boat at time \(t ;\) (b) the limiting velocity (the velocity after a long time has passed).

A small space probe is released from a spaceship. The space probe has mass \(20.0 \mathrm{kg}\) and contains \(90.0 \mathrm{kg}\) of fuel. It starts from rest in deep space, from the origin of a coordinate system based on the spaceship, and burns fuel at the rate of \(3.00 \mathrm{kg} / \mathrm{s}\). The engine provides a constant thrust of \(120.0 \mathrm{N}\). (a) Write an expression for the mass of the space probe as a function of time, between 0 and 30 seconds, assuming that the engine ignites fuel beginning at \(t=0 .\) (b) What is the velocity after \(15.0 \mathrm{s}\) ? (c) What is the position of the space probe after \(15.0 \mathrm{s}\), with initial position at the origin? (d) Write an expression for the position as a function of time, for \(t>30.0 \mathrm{s}\)

(a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked \(50.0 \mathrm{m}\) radius curve at \(30.0 \mathrm{m} / \mathrm{s}\). (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?

An elevator filled with passengers has a mass of \(1.70 \times 10^{3} \mathrm{kg} .\) (a) The elevator accelerates upward from rest at a rate of \(1.20 \mathrm{m} / \mathrm{s}^{2}\) for \(1.50 \mathrm{s}\). Calculate the tension in the cable supporting the elevator. (b) The elevator continues upward at constant velocity for 8.50 s. What is the tension in the cable during this time? (c) The elevator decelerates at a rate of \(0.600 \mathrm{m} / \mathrm{s}^{2}\) for \(3.00 \mathrm{s}\). What is the tension in the cable during deceleration? (d) How high has the elevator moved above its original starting point, and what is its final velocity?
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