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Step-by-Step Solution
Step 1: Identify the Given Quantities
• Mass of the body, m = 2\text{ kg}
• Force acting on the body, \vec{F} = 2\hat{i} + 3\hat{j} + 5\hat{k}\text{ N}
• Time interval, t = 4\text{ s}
• Initial position, \vec{r_i} = 0\hat{i} + 0\hat{j} + 0\hat{k}
• Initial velocity, \vec{u} = 0\hat{i} + 0\hat{j} + 0\hat{k} (starts from rest)
• Final position after 4 s is given as (x,\, b,\, z) = (8,\, b,\, 20) .
Step 2: Calculate the Acceleration
According to Newton's Second Law,
\vec{F} = m\vec{a}.
Hence, the acceleration of the body is
\vec{a} = \frac{\vec{F}}{m} = \frac{2\hat{i} + 3\hat{j} + 5\hat{k}}{2} = 1\hat{i} + 1.5\hat{j} + 2.5\hat{k}\,\text{m/s}^2.
Step 3: Use the Equation of Motion for Constant Acceleration
Because the initial velocity \vec{u} = 0 , the displacement after time t under constant acceleration \vec{a} is given by
\vec{s} = \vec{u} t + \frac{1}{2}\vec{a}t^2.
Substituting \vec{u} = 0 and \vec{a} = (1\hat{i} + 1.5\hat{j} + 2.5\hat{k}) , we get
\vec{s} = \frac{1}{2}(1\hat{i} + 1.5\hat{j} + 2.5\hat{k})(4^2).
Step 4: Compute the Displacement Vector
Since t = 4\,\text{s} ,
\vec{s} = \frac{1}{2}\,(1\hat{i} + 1.5\hat{j} + 2.5\hat{k}) \times 16.
\vec{s} = 8\,(1\hat{i} + 1.5\hat{j} + 2.5\hat{k}).
So,
\vec{s} = (8 \times 1)\hat{i} + (8 \times 1.5)\hat{j} + (8 \times 2.5)\hat{k} = (8\hat{i} + 12\hat{j} + 20\hat{k}).
Step 5: Compare with the Final Position
The final coordinates are given as (8,\, b,\, 20) from the question. From the displacement calculation,
x = 8,\quad y = 12,\quad z = 20.
Hence, b = 12.
Final Answer
The value of b is 12.
