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The velocity v of a moving particle changes with displacement x as follows v = 2x 1 . Find, how does x vary with time t. (Assume initial position to be at x = 0 ) Ans is x = 1/2(e^2t - 1) Can anyone explain ?
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The velocity v of a moving particle changes with displacement x as fol...
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The velocity v of a moving particle changes with displacement x as fol...
v = 2x + 1
v = dx/dt
=> dx/dt = 2x + 1
=> dx = (2x+1)dt
=> dt = dx/(2x + 1)
Integrate both sides
=> ∫ dt = ∫(1/(2x+1)) dx
=> t = ln | 2x + 1| / 2 + C
at t = 0 x = 0
=> 0 = ln | 2(0) + 1| / 2 + C
=> 0 = ln | 1| / 2 + C
=> 0 =0 / 2 + C
=> C = 0
t = ln | 2x + 1| / 2
=> 2t = ln | 2x + 1|
=> 2x + 1 = 
=> x = ( 
- 1)/2
This is how x varies with time t
Community Answer
The velocity v of a moving particle changes with displacement x as fol...
Understanding the Problem:
Given v = 2x - 1, we need to find how x varies with time t when the initial position is at x = 0.
Relation between Velocity and Displacement:
The velocity v is the rate of change of displacement x with respect to time t. Mathematically, v = dx/dt.
Initial Condition:
Given x = 0 at t = 0, we can find the value of the constant of integration.
Integration to find x:
We integrate v = 2x - 1 with respect to x to find x in terms of t.
Integrating v with respect to x:
∫v dx = ∫(2x - 1) dx
=> x^2 - x = ∫v dx
Substitute for v:
Since v = dx/dt, we can rewrite the above equation in terms of t.
Integrating with respect to t:
∫(dx/dt) dt = x^2 - x
=> x = x(t) = 1/2(e^(2t) - 1)
Conclusion:
Therefore, the variation of x with time t is given by x = 1/2(e^(2t) - 1).
Given v = 2x - 1, we need to find how x varies with time t when the initial position is at x = 0.
Relation between Velocity and Displacement:
The velocity v is the rate of change of displacement x with respect to time t. Mathematically, v = dx/dt.
Initial Condition:
Given x = 0 at t = 0, we can find the value of the constant of integration.
Integration to find x:
We integrate v = 2x - 1 with respect to x to find x in terms of t.
Integrating v with respect to x:
∫v dx = ∫(2x - 1) dx
=> x^2 - x = ∫v dx
Substitute for v:
Since v = dx/dt, we can rewrite the above equation in terms of t.
Integrating with respect to t:
∫(dx/dt) dt = x^2 - x
=> x = x(t) = 1/2(e^(2t) - 1)
Conclusion:
Therefore, the variation of x with time t is given by x = 1/2(e^(2t) - 1).
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The velocity v of a moving particle changes with displacement x as follows v = 2x 1 . Find, how does x vary with time t. (Assume initial position to be at x = 0 ) Ans is x = 1/2(e^2t - 1) Can anyone explain ?
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The velocity v of a moving particle changes with displacement x as follows v = 2x 1 . Find, how does x vary with time t. (Assume initial position to be at x = 0 ) Ans is x = 1/2(e^2t - 1) Can anyone explain ? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about The velocity v of a moving particle changes with displacement x as follows v = 2x 1 . Find, how does x vary with time t. (Assume initial position to be at x = 0 ) Ans is x = 1/2(e^2t - 1) Can anyone explain ? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The velocity v of a moving particle changes with displacement x as follows v = 2x 1 . Find, how does x vary with time t. (Assume initial position to be at x = 0 ) Ans is x = 1/2(e^2t - 1) Can anyone explain ?.
The velocity v of a moving particle changes with displacement x as follows v = 2x 1 . Find, how does x vary with time t. (Assume initial position to be at x = 0 ) Ans is x = 1/2(e^2t - 1) Can anyone explain ? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about The velocity v of a moving particle changes with displacement x as follows v = 2x 1 . Find, how does x vary with time t. (Assume initial position to be at x = 0 ) Ans is x = 1/2(e^2t - 1) Can anyone explain ? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The velocity v of a moving particle changes with displacement x as follows v = 2x 1 . Find, how does x vary with time t. (Assume initial position to be at x = 0 ) Ans is x = 1/2(e^2t - 1) Can anyone explain ?.
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