18) A particle moves in a straight line it begins its motios from a fixed point \( (O) \) on the straigh line such that the algebraic measure of its acceleration is given in terms of the algebraic measure of its position by the relation \( a=2 \chi+5 \). Given that the initial velocity of the particle is \( 2 \mathrm{~m} / \mathrm{sec} \)., then \( \chi= \) \( \qquad \) m. at \( \mathrm{v}=4 \mathrm{~m} / \mathrm{sec} \). (a) 1 or -6 (b) -1 or 6 (C) -1 (d) 6 19)A particle moves in a straight line from a fixed point on the straight line starting at rest such that \( \mathrm{a}=\frac{3}{8} x^{2} \) where a is measured in \( \mathrm{m} / \mathrm{sec}^{2} \). and \( x \) in metre, then the velocity of the particle when \( X=2 \) metres \( = \) \( \qquad \) \( \mathrm{m} / \mathrm{sec} \). (a) \( \sqrt{2} \) (b) 2 (c) \( \sqrt{3} \) (d) 3 20) A particle moves in a straight line with initial speed \( 2 \mathrm{~m} / \mathrm{sec} \). from a fixed point ( 0 ) on a straight line such that \( \mathrm{a}=\mathrm{e}^{x} \), then \( \mathrm{v}^{2}= \) \( \qquad \) (a) \( 2 e^{x} \) (b) \( 2 e^{x}+2 \) (c) \( 3 e^{x}-1 \) (d) \( 2 e^{x}+1 \) 21) A particle moves in a straight line with initial speed \( 8 \mathrm{~m} / \mathrm{sec} \). from a fixed point ( 0 ) on the straight line such that \( a=40 e^{-x} \), then the maximum speed of the particle \( = \) \( \qquad \) \( \mathrm{m} / \mathrm{sec} \). (a) 12 (b) 80 (C) 144 (d) \( \ln 80 \)
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Bonus Knowledge
Let's jazz things up with a little history lesson and a dash of real-world application! Did you know that the relationship between acceleration and position, like in these particle motion problems, can be traced back to Sir Isaac Newton? His second law, \( F = ma \), laid the groundwork for classical mechanics, helping us understand how forces affect motion. This foundational concept has powered everything from ancient chariots to modern space shuttles. Now, vectors aren't just for navigation! Understanding particle motion helps engineers design safer vehicles, optimize roller coasters, and even launch satellites. When testing new designs, physics principles guide how these moving objects will perform under different forces, making the thrill or convenience of your ride depend on some serious math.
