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A particle has initial velocity $(3\hat{i}+4\hat{j})$ and acceleration $(0.4\hat{i}+0.3\hat{j})$. Its speed after 10 s is:
Velocity at $t = 10\text{ s}$:
$\vec{v} = (3+0.4\times10)\hat{i} + (4+0.3\times10)\hat{j} = 7\hat{i}+7\hat{j}$
Speed $= \sqrt{7^2+7^2} = \sqrt{98} = \mathbf{7\sqrt{2}}$ units
From a collection of 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are chosen and arranged in a row such that the dictionary always occupies the middle position. How many such arrangements exist?
Choose 4 novels from 6: $\binom{6}{4} = 15$ ways.
Choose 1 dictionary from 3: $\binom{3}{1} = 3$ ways.
Arrange 5 books in a row with dictionary fixed in the middle: the 4 novels occupy the 4 remaining positions, giving $4! = 24$ arrangements.
Total $= 15 \times 3 \times 24 = \mathbf{1080}$
Since $1080 \geq 1000$, the answer is at least 1000.
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