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Chapter 8: Problem 52

A rain drop starts falling from a height of \(2 \mathrm{~km}\). It falls with a continuously decreasing acceleration and attains its terminal velocity at a height of \(1 \mathrm{~km}\). The ratio of the work done by the gravitational force in the first half and the second half of the drops journey is (1) \(1: 1\) and the times of fall of the drop in the two halves is \(a: 1\) (where \(a>1\) ) (2) \(1: 1\) and the times of fall of the drop in the two halves is \(a: 1\) (where \(a<1\) ) (3) \(a: 1\) (where \(a>1\) ) and the times of fall of the drop in the two halves is \(1: 1\) (4) \(a: 1\) (where \(a<1\) ). and the times of fall of the drop in the two halves is \(1: 1\)

Short Answer

Expert verified
Option (1) matches: work done is in ratio 1:1 and times are in ratio \(a:1\) with \(a>1\).

Step by step solution

01

Determine the Work Done Formula

The work done by the gravitational force on an object falling from height is given by the formula \( W = mgh \) where \( m \) is the mass of the object, \( g \) is the gravitational acceleration, and \( h \) is the height through which the object moves.
02

Calculate Work Done in Each Half

The raindrop falls from a height of \( 2 \mathrm{~km} \) to \( 1 \mathrm{~km} \), thus covering a height of \( 1 \mathrm{~km} \) in each half. Since the height covered is the same in both halves and gravitational acceleration \( g \) remains constant, the work done is the same for both halves: \( W_1 = W_2 = mg \times 1 \mathrm{~km} \).
03

Conclusion about Work Done

Since \( W_1 = W_2 \), the ratio of work done in both halves is \( 1:1 \).
04

Consider Acceleration

The drop falls with decreasing acceleration until it reaches terminal velocity at \( 1 \mathrm{~km} \). This means it takes more time to cover the first half than the second because it accelerates continuously during this period but reaches constant speed in the second half.
05

Determine Time Ratio

Since the drop has decreasing acceleration in the first half and is faster due to terminal velocity in the second half, the time taken in the first half is greater than in the second half. Thus, the time ratio is \( a:1 \) where \( a > 1 \).
06

Find the Correct Option

From the ratio of works \( 1:1 \) and time \( a:1 \) \((a>1)\), option (1) matches exactly.

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Key Concepts

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

Work-Energy Theorem
The work-energy theorem states that the work done by all forces acting on an object equals the change in its kinetic energy. When it comes to falling objects like the raindrop in our exercise, the primary force is gravity. As the raindrop falls, gravity does work on it, converting gravitational potential energy into kinetic energy. To understand this better:
  • At the start of its journey, the raindrop has high gravitational potential energy and no kinetic energy.
  • As it falls, potential energy decreases while kinetic energy increases as it speeds up.
  • When it reaches terminal velocity, its speed becomes constant, meaning no further acceleration occurs and no more conversion between energy types due to gravity alone.
In our exercise, because the height across which the raindrop travels is the same in both halves of its journey, and gravitational force remains constant, the work-energy theorem tells us the work done by gravity is equal in both halves. This is why the ratio of work done is 1:1.
Terminal Velocity
Terminal velocity is the constant speed that a free-falling object eventually reaches. It occurs when the force of gravity pulling the object down is balanced by the drag force (air resistance) pushing it upward. For the raindrop:
  • Initially, when it starts falling, it accelerates because gravity pulls it down faster than air resistance can slow it down.
  • As its speed increases, so does air resistance until it grows strong enough to balance out the gravitational pull.
  • At this point, the raindrop reaches terminal velocity and falls at a steady speed of zero acceleration.
In the exercise, the raindrop reaches terminal velocity halfway through its descent, meaning the first half involves increasing speed and the second half involves constant speed. The time taken to reach terminal velocity requires more effort (and therefore time) than maintaining it, explaining the longer time spent in the first half of the descent.
Rain Drop Motion
Raindrop motion is influenced by several forces: gravity, air resistance, and sometimes wind. Understanding its motion involves looking at how these forces interact over the course of its fall. In the problem at hand:
  • The raindrop starts from a static position at 2 km and falls to 1 km with decreasing acceleration due to growing air resistance.
  • By 1 km, it achieves terminal velocity, a point where gravitational force and air resistance are equal. Thus, it transitions from varying speed to steady motion.
  • This transition affects the timing of the descent. It initially takes more time to accelerate and combat air resistance before terminal velocity is reached.
The exercise emphasizes that as the raindrop continues after reaching terminal velocity, it completes the second half faster than the first, leading to unequal time intervals despite equal distance.

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Most popular questions from this chapter

A capillary tube is attached horizontally to a constant pressure head arrangement. If the radius of the capillary tube is increased by \(10 \%\), then the rate of flow of the liquid, shall change nearly by (1) \(+10 \%\) (2) \(46 \%\) (3) \(-10 \%\) (4) \(-40 \%\)

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