14) A body starts its motion from rest from a fixed point \( (O) \) in a horizontal straight line tvith acceleration \( a=(12-61) \mathrm{cm} / \mathrm{sec}^{2} \), where the time \( t \) in seconds , so the distance betiveen the body and the point \( (O) \) when it stops instantly \( = \). \( \qquad \) cm. (a) 16 (b) 32 (c) 64 (d) 96 15) Tivo bodies began their motions in a straight line from the same point and at the same time and in the same direction, one of them moves with velocity \( v_{1}=\left(3 t^{2}+2 t\right) \mathrm{m} / \mathrm{sec} \), and the other moves ivith velocity \( v_{2}=(21) \mathrm{m} / \mathrm{sec} \)., then the distance berween them after 6 seconds from the beginning \( = \) \( \qquad \) m. (a) 6 (b) 36 (C) 108 (d) 216 16) Aody moves in a straight line with velocity \( \mathrm{v}=(6-2 \mathrm{t}) \mathrm{cm} \). \( / \mathrm{sec} \)., the body began its motion 3 cm . on the right side of a fixed point \( (O) \) on the straight line then : First: The position of the body after 2 seconds from the beginning of motion = \( \qquad \) cm. (a) 8 (b) 11 (c) 12 (d) 19 Second : The displacement in the time interval \( [1,5] \) equals (a) -10 (b) zero (C) 10 (d) 20 Third: The total distance in the time interval \( [1,5] \) equals (a) zero (b) 4 (c) 8 (d) 12 17) A particle moves in a straight line from a fixed point (O) with initial speed \( 10 \mathrm{~m} / \mathrm{sec} \). such that the algebraic measure of its acceleration is given in term of the algebraic measure of its position s by the relation \( a=2 x+3 \), then its speed at \( x=14 \mathrm{~m} \), equals \( \qquad \) \( \mathrm{m} / \mathrm{sec} \). (a) 24 (b) 34 (c) 476 (d) 576
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Initially, using the formula of motion with constant acceleration, we can find the distance the body covers before it stops. Its acceleration, given as \( a = (12 - 6t) \text{ cm/sec}^2 \), will eventually result in a zero velocity, indicating the distance can be calculated by integrating the acceleration function until the velocity becomes zero, arriving at the correct option of 64 cm. Now, for the next question about two bodies moving from the same point, we can compute their respective positions at 6 seconds. The first body has a non-linear velocity equation, creating a uniquely different position from the constant velocity of the second body. By integrating the velocity of the first body from 0 to 6 seconds and subtracting the second body's distance, we reveal that the distance between them is indeed 108 meters.
