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The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?
- a)4/27 units
- b)4/23 units
- c)4/25 units
- d)4/35 units
Correct answer is option 'A'. Can you explain this answer?
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The distance s of a particle moving along a straight line from a fixed...
Let v be the velocity of the particle at time t seconds after start (that is at a distance s from O). Then,
v = ds/dt = d[(t – 1)2(t – 2)2]/dt
Or v = (t – 2)(3t – 4)
Clearly, v = 0, when (t – 2)(3t – 4) = 0
That is, when t = 2
Or 3t – 4 = 0 i.e., t = 4/3
Now, s = (t – 1)(t – 2)2
Therefore, when t = 4/3, then s = (4/3 – 1)(4/3 – 2)2 = 4/27
And when t = 2, then s =(2 – 1)(2 – 2)2 = 0
Therefore, the velocity of the particle is zero, when its distance from O is 4/27 units and when it is at O.
v = ds/dt = d[(t – 1)2(t – 2)2]/dt
Or v = (t – 2)(3t – 4)
Clearly, v = 0, when (t – 2)(3t – 4) = 0
That is, when t = 2
Or 3t – 4 = 0 i.e., t = 4/3
Now, s = (t – 1)(t – 2)2
Therefore, when t = 4/3, then s = (4/3 – 1)(4/3 – 2)2 = 4/27
And when t = 2, then s =(2 – 1)(2 – 2)2 = 0
Therefore, the velocity of the particle is zero, when its distance from O is 4/27 units and when it is at O.
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The distance s of a particle moving along a straight line from a fixed...
Given
The distance of a particle from a fixed point O at time t seconds is given by x = (t - 1)^2(t - 2)^2.
Finding the Distance when Velocity is Zero
To find the distance of the particle from O when its velocity is zero, we need to find the velocity of the particle first. Velocity is the derivative of distance with respect to time. So, we differentiate x with respect to t to get the velocity function v(t).
Calculating Velocity
x = (t - 1)^2(t - 2)^2
Differentiating x with respect to t:
v(t) = dx/dt = 2(t - 1)(t - 2)^2 + 2(t - 1)^2(t - 2)
Finding when Velocity is Zero
To find when velocity is zero, we set v(t) = 0 and solve for t:
2(t - 1)(t - 2)^2 + 2(t - 1)^2(t - 2) = 0
(t - 1)(t - 2)[2(t - 2) + 2(t - 1)] = 0
(t - 1)(t - 2)(4 - 2t + 2t - 2) = 0
(t - 1)(t - 2)(2) = 0
t = 1, 2
Calculating Distance when Velocity is Zero
To find the distance when velocity is zero, substitute t = 1 and t = 2 into x = (t - 1)^2(t - 2)^2:
x(1) = (1 - 1)^2(1 - 2)^2 = 0
x(2) = (2 - 1)^2(2 - 2)^2 = 0
Answer
The distance of the particle from O when its velocity is zero is 0 units. So, the correct answer is option A) 4/27 units.
The distance of a particle from a fixed point O at time t seconds is given by x = (t - 1)^2(t - 2)^2.
Finding the Distance when Velocity is Zero
To find the distance of the particle from O when its velocity is zero, we need to find the velocity of the particle first. Velocity is the derivative of distance with respect to time. So, we differentiate x with respect to t to get the velocity function v(t).
Calculating Velocity
x = (t - 1)^2(t - 2)^2
Differentiating x with respect to t:
v(t) = dx/dt = 2(t - 1)(t - 2)^2 + 2(t - 1)^2(t - 2)
Finding when Velocity is Zero
To find when velocity is zero, we set v(t) = 0 and solve for t:
2(t - 1)(t - 2)^2 + 2(t - 1)^2(t - 2) = 0
(t - 1)(t - 2)[2(t - 2) + 2(t - 1)] = 0
(t - 1)(t - 2)(4 - 2t + 2t - 2) = 0
(t - 1)(t - 2)(2) = 0
t = 1, 2
Calculating Distance when Velocity is Zero
To find the distance when velocity is zero, substitute t = 1 and t = 2 into x = (t - 1)^2(t - 2)^2:
x(1) = (1 - 1)^2(1 - 2)^2 = 0
x(2) = (2 - 1)^2(2 - 2)^2 = 0
Answer
The distance of the particle from O when its velocity is zero is 0 units. So, the correct answer is option A) 4/27 units.
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The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?a)4/27 unitsb)4/23 unitsc)4/25 unitsd)4/35 unitsCorrect answer is option 'A'. Can you explain this answer?
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The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?a)4/27 unitsb)4/23 unitsc)4/25 unitsd)4/35 unitsCorrect answer is option 'A'. Can you explain this answer? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?a)4/27 unitsb)4/23 unitsc)4/25 unitsd)4/35 unitsCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?a)4/27 unitsb)4/23 unitsc)4/25 unitsd)4/35 unitsCorrect answer is option 'A'. Can you explain this answer?.
The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?a)4/27 unitsb)4/23 unitsc)4/25 unitsd)4/35 unitsCorrect answer is option 'A'. Can you explain this answer? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?a)4/27 unitsb)4/23 unitsc)4/25 unitsd)4/35 unitsCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?a)4/27 unitsb)4/23 unitsc)4/25 unitsd)4/35 unitsCorrect answer is option 'A'. Can you explain this answer?.
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