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Chapter 2: Problem 57

A parachutist whose total mass is \(100 \mathrm{kg}\) is falling at \(50 \mathrm{m} / \mathrm{s}\) when her parachute opens. Her speed drops to \(6 \mathrm{m} / \mathrm{s}\) in 2 s. What is the total force her harness had to withstand? How many times her weight is this force?

Short Answer

Expert verified
The total force is 2200 N. The harness force is approximately 2.24 times her weight.

Step by step solution

01

Identify Given Information

The total mass of the parachutist is given as 100 kg, her initial velocity is 50 m/s, her final velocity is 6 m/s, and the time taken for the speed to drop is 2 seconds.
02

Calculate Acceleration

Use the formula for acceleration, which is given by \( a = \frac{v_f - v_i}{t} \), where \(v_f\) is the final velocity, \(v_i\) is the initial velocity, and \(t\) is the time. Substitute the given values: \( a = \frac{6 - 50}{2} = \frac{-44}{2} = -22 \, \mathrm{m/s^2} \).
03

Calculate Total Force

Use Newton's Second Law of Motion, \( F = ma \), where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration. Substitute the values: \( F = 100 \, \mathrm{kg} \times -22 \, \mathrm{m/s^2} = -2200 \, \mathrm{N} \). The negative sign indicates that the force direction is opposite to the direction of motion.
04

Calculate Weight

The weight of the parachutist is given by \( W = mg \), where \( g \) is the acceleration due to gravity \((9.8 \, \mathrm{m/s^2}\)). Substitute the values: \( W = 100 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2} = 980 \, \mathrm{N} \).
05

Calculate Force in Terms of Weight

To find how many times the harness force is greater than her weight, divide the total force by the weight: \( \frac{|F|}{W} = \frac{2200 \, \mathrm{N}}{980 \, \mathrm{N}} = \frac{2200}{980} \approx 2.24 \).The harness force is approximately 2.24 times her weight.

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Key Concepts

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

Newton's Second Law
Newton's Second Law of Motion is fundamental in mechanics. This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, it can be expressed as:

\ F = ma \

Here, \( F \) is the net force applied to the object, \( m \) is the mass of the object, and \( a \) is the acceleration. This means that to change the velocity of an object (i.e., to accelerate it), a force must be applied. The greater the mass of the object, the more force is required to achieve the same acceleration.

In the given exercise, this principle is used to determine the total force acting on the parachutist when her speed reduces drastically upon the deployment of the parachute.
Acceleration
Acceleration describes the rate of change of velocity of an object. It鈥檚 given by the formula:

\ a = \frac{v_f - v_i}{t} \

where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, and \( t \) is the time over which this change occurs.

For the parachutist problem, we calculate:

\ a = \frac{6 - 50}{2} = -22 \, \mathrm{m/s^2} \

The negative acceleration indicates a deceleration, meaning her speed reduces due to the parachute opening. Understanding the concept of acceleration is crucial because it links the change in speed with the forces applied, as mandated by Newton鈥檚 Second Law.
Weight and Force
Weight and force are often confused, but they are distinct concepts. Weight is the force exerted by gravity on an object鈥檚 mass. It can be calculated using:

\ W = mg \

where \( W \) is the weight, \( m \) is the mass, and \( g \) is the acceleration due to gravity (9.8 \, m/s^2).

For a parachutist with a mass of 100 kg, her weight is:

\ W = 100 \, kg \times 9.8 \, m/s^2 = 980 \, N \

Force, on the other hand, is the interaction that changes the motion of an object. When the parachute opens, the harness has to withstand a force due to rapid deceleration:

\ F = ma = 100 \, kg \times -22 \, m/s^2 = -2200 \, N \

The parachutist experiences a force 2.24 times her weight because the total force (2200 N) divided by her weight (980 N) gives approximately 2.24. This considerable force is why parachutes and harnesses need to be extremely robust.

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Most popular questions from this chapter

A man is rowing at \(8 \mathrm{km} / \mathrm{h}\) in a river \(1.5 \mathrm{km}\) wide in which the current is \(5 \mathrm{km} / \mathrm{h}\). (a) In what direction should he head in order to get across the river in the shortest possible time? (b) How much time will he take if he goes in this direction? (c) How far downstream will the boat have gone when it reaches the opposite side?

(a) Imagine that Charlotte drops a ball from a window on the twentieth floor of a building while at the same time Fred drops another ball from a window on the nineteenth floor of that building. As the balls fall, what happens to the distance between them (assuming no air resistance)? (b) Next imagine that Charlotte and Fred are at the same window on the twentieth floor and that Fred drops his ball a few seconds after Charlotte drops hers. As the balls fall, what happens to the distance between them now (again assuming no air resistance)?

A force of \(20 \mathrm{N}\) gives an object an acceleration of \(5 \mathrm{m} / \mathrm{s}^{2} .\) (a) What force would be needed to give the same object an acceleration of \(1 \mathrm{m} / \mathrm{s}^{2} ?\) (b) What force would be needed to give it an acceleration of \(10 \mathrm{m} / \mathrm{s}^{2} ?\)

A woman jogs for \(2 \mathrm{km}\) at \(8 \mathrm{km} / \mathrm{h}\) and then walks for \(2 \mathrm{km}\) at \(6 \mathrm{km} / \mathrm{h}\). What is her average speed for the entire trip?

The acceleration of a certain moving object is constant in magnitude and direction. Must the path of the object be a straight line? If not, give an example.
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