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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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