A particle thrown up has zero velocity but how much will be the acceleration ?

Question

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Correct — option (3), equal to acceleration due to gravity. The question describes the instant at the very top of a vertical throw, where the particle has stopped rising and has not yet begun to fall, so its velocity is momentarily zero. Its acceleration at that instant is not zero: it is g, the same acceleration it had on the way up and will have on the way down, directed vertically downward. The reason is Newton's second law. Once the particle has left the hand there is only one force acting on it, its own weight, and if air resistance is neglected that force is constant throughout the flight; a constant force on a constant mass produces a constant acceleration, so nothing about the acceleration changes at the top or anywhere else. The confusion the question is testing comes from treating velocity and acceleration as if one implied the other. Acceleration is the rate at which velocity changes, not the amount of velocity present, so a body can have zero velocity while its velocity is changing as fast as it ever does. At the top of the throw the velocity is passing through zero on its way from upward to downward, and it is passing through at a rate of about 9.8 metres per second every second — which is why the particle does not hover there but starts falling immediately. Indeed if the acceleration really were zero at that point, a particle with zero velocity and zero acceleration would simply stay where it was and never come down. The same pattern appears at the extreme position of a swinging pendulum or a body in simple harmonic motion, where the displacement is greatest, the velocity is zero and the acceleration is at its maximum.

Why the other options are wrong — (a) Zero is the intended trap and the answer that intuition supplies, because the particle does appear to be at rest for an instant and rest is naturally associated with nothing happening. But acceleration measures how fast velocity is changing, not how much of it there is, and at the top of the flight the velocity is changing sign — from upward to downward — which is a change at full rate rather than no change at all. The decisive check is to ask what would follow if the acceleration were genuinely zero: a body with zero velocity and zero acceleration remains at rest permanently, so the particle would hang at the top of its flight and never return, which is plainly not what happens.  (b) Infinity would require an infinite force, since acceleration is force divided by mass and the mass of the particle is finite and unchanging. The only force acting after release is the weight of the particle, which is its mass multiplied by g and is therefore perfectly finite; no new force appears at the top of the trajectory, and nothing in the motion is discontinuous. Infinite acceleration would also mean an infinite change of velocity in an instant, whereas the particle's speed changes smoothly and gradually through zero as it turns around.  (d) One is not an answer at all until it is given a unit, and no system of units makes the acceleration of a freely falling body equal to one. In SI units the acceleration due to gravity near the earth's surface is about 9.8 metres per second squared, so a bare numeral of one is wrong by nearly a factor of ten even if metres per second squared were assumed. This choice is worth noticing as a reminder that a physical quantity without its unit is meaningless, and that an option offering a naked number in a question about acceleration can be rejected on that ground alone.

Concept: Velocity and acceleration are independent quantities at any given instant, and the vertical throw is the standard demonstration of it. Take upward as positive and let the particle leave the hand with speed u. Its velocity at time t is u minus gt, which falls steadily, passes through zero at time u divided by g, and then becomes negative as the particle descends. Its acceleration is minus g at every instant of that history, constant in both magnitude and direction, because the only force acting is the weight. From these two relations the whole motion follows: the greatest height reached is u squared divided by twice g, the time to rise equals the time to fall, and the particle returns to the thrower's hand with the same speed it left, though with the opposite direction. Galileo's insight that this acceleration is the same for all bodies regardless of mass, once air resistance is removed, is what makes g a property of the earth rather than of the object. The identical logic governs the extreme point of any oscillation: at the ends of the swing of a pendulum or of a body in simple harmonic motion the velocity is zero while the restoring force, and therefore the acceleration, is at its greatest.

State PCS 2024 Q51 — Science & Technology PYQ with Answer

State PCS 2024 Q51 — Science & Technology PYQ with Answer