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A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?
- a)The speed of the raindrop increases linearly with time.
- b)The speed of the raindrop increases exponentially with time.
- c)The speed of the raindrop approaches a constant value when λt >>1.
- d)The speed of the raindrop approaches a constant value when λt << 1.
Correct answer is option 'C'. Can you explain this answer?
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A raindrop falls under gravity and captures water molecules from atmos...
Proportional to the surface area of the raindrop. This can be described by the following differential equation:
dm/dt = kA
where m is the mass of the raindrop, t is time, A is the surface area of the raindrop, and k is a constant of proportionality.
As the raindrop falls, it gains more and more water molecules from the atmosphere, causing its mass to increase. However, as the raindrop gets bigger, its surface area also increases, which means it captures water molecules at a faster rate.
The differential equation above can be solved using separation of variables:
dm/A = k dt
Integrating both sides:
ln(A) = kt + C
where C is the constant of integration.
Exponentiating both sides:
A = e^(kt+C) = C'e^(kt)
where C' is a new constant of integration that combines C and e^C.
This equation shows that the surface area of the raindrop increases exponentially with time, which is why raindrops can grow very quickly as they fall through the atmosphere.
dm/dt = kA
where m is the mass of the raindrop, t is time, A is the surface area of the raindrop, and k is a constant of proportionality.
As the raindrop falls, it gains more and more water molecules from the atmosphere, causing its mass to increase. However, as the raindrop gets bigger, its surface area also increases, which means it captures water molecules at a faster rate.
The differential equation above can be solved using separation of variables:
dm/A = k dt
Integrating both sides:
ln(A) = kt + C
where C is the constant of integration.
Exponentiating both sides:
A = e^(kt+C) = C'e^(kt)
where C' is a new constant of integration that combines C and e^C.
This equation shows that the surface area of the raindrop increases exponentially with time, which is why raindrops can grow very quickly as they fall through the atmosphere.
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A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer?
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A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer?.
A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer?.
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A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer?, a detailed solution for A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?a)The speed of the raindrop increases linearly with time.b)The speed of the raindrop increases exponentially with time.c)The speed of the raindrop approaches a constant value when λt >>1.d)The speed of the raindrop approaches a constant value when λt << 1.Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice IIT JAM tests.
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