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Step-by-Step Solution
Step 1: Identify the given data
• Mass of the object, m = 100\,\text{g} = 0.1\,\text{kg}
• Force acting on the object, \overrightarrow{F} = (10\,\hat{i} + 5\,\hat{j})\,\text{N}
• Time after which position is required, t = 2\,\text{s}
• Initial velocity of the object is zero (starts from rest).
Step 2: Apply Newton’s Second Law to find acceleration
According to Newton’s Second Law, \overrightarrow{F} = m\,\overrightarrow{a} .
Therefore,
\overrightarrow{a} = \frac{\overrightarrow{F}}{m}
= \frac{10\,\hat{i} + 5\,\hat{j}}{0.1}
= 100\,\hat{i} + 50\,\hat{j}\,\text{m/s}^2.
Step 3: Use kinematics to find displacement after 2 s
The object starts from rest, so the displacement after time t under constant acceleration is given by:
\overrightarrow{S} = \frac{1}{2}\,\overrightarrow{a}\,t^2.
Substituting \overrightarrow{a} = (100\,\hat{i} + 50\,\hat{j}) and t = 2\,\text{s} , we get:
\overrightarrow{S}
= \frac{1}{2} \times (100\,\hat{i} + 50\,\hat{j}) \times (2)^2
= \frac{1}{2} \times (100\,\hat{i} + 50\,\hat{j}) \times 4
= 2 \times (100\,\hat{i} + 50\,\hat{j})
= 200\,\hat{i} + 100\,\hat{j}\,\text{m}.
Hence, a = 200\,\text{m} and b = 100\,\text{m} .
Step 4: Find the ratio a/b
We have:
\frac{a}{b} = \frac{200}{100} = 2.
Thus, the required ratio \frac{a}{b} is 2.
