MPSC 2024 Prelims Paper-I · Question 51 of 100

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.

• (1)Zero — 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.
• (2)Infinity — 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.
• (4)One — 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.

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.

MPSC's physics questions are conceptual rather than computational, and the ones it repeats most are those where everyday intuition points the wrong way — velocity zero but acceleration not zero, weightlessness in orbit despite gravity acting, a body moving uniformly in a circle being accelerated even though its speed is constant. The habit that answers all of them is to go back to the forces: identify every force acting at the instant in question, sum them, and divide by the mass. If the sum of the forces is not zero then the acceleration is not zero, whatever the velocity happens to be doing. That check takes a few seconds in the examination hall and is immune to the intuition the question is designed to exploit.

• The acceleration due to gravity near the earth's surface is about 9.8 metres per second squared, directed vertically downward, and it is the same for all bodies irrespective of their mass once air resistance is neglected.
• At the highest point of a vertical throw the velocity is momentarily zero but the acceleration is unchanged at g, because the only force acting on the particle after release is its weight.
• Acceleration is the rate of change of velocity, so a body can have zero velocity and non-zero acceleration; a body with both zero would remain permanently at rest.
• For a particle thrown upward with initial speed u, the maximum height reached is u squared divided by twice g, the time of ascent equals the time of descent, and it returns with the same speed it started with.
• The same relation appears at the extreme positions of simple harmonic motion, where displacement and acceleration are greatest while the velocity passes through zero.

• Inferring zero acceleration from zero velocity, when the two are independent and only the forces determine the acceleration
• Forgetting that acceleration is a vector, so that a particle turning around at the top of its flight is undergoing a change of velocity at full rate
• Accepting a bare number as an answer to a question about a physical quantity, when without its unit it states nothing
• Assuming uniform circular motion is unaccelerated because the speed is constant, which is the same confusion in a different setting

MPSC sets mechanics questions as one-line conceptual traps rather than as numerical problems: the highest point of a throw, the bob at the end of its swing, a satellite in orbit, a lift accelerating downward. Each is answered by returning to the forces acting at the instant described, and each has one option that rewards intuition rather than physics. Expect this particular item's theme to reappear either as a comparison of velocity and acceleration at the top of a trajectory, or as a question on simple harmonic motion asking where displacement, velocity and acceleration take their maximum and minimum values.

No directly related past PYQ was found.

• practice — not a real PYQ

  A body performing simple harmonic motion is at its extreme position. Which of the following is correct at that instant ?

  • (a)Both velocity and acceleration are zero
  • (b)Velocity is zero and acceleration is maximum
  • (c)Velocity is maximum and acceleration is zero
  • (d)Both velocity and acceleration are maximum

  Answer(b) Velocity is zero and acceleration is maximum — at the extreme position the displacement from the mean position is greatest, so the restoring force and hence the acceleration are greatest, while the body has momentarily stopped before reversing. The opposite holds at the mean position, where the velocity is maximum and the acceleration is zero.

• practice — not a real PYQ

  A ball is thrown vertically upward with an initial speed of 19.6 metres per second. Taking g as 9.8 metres per second squared and neglecting air resistance, the maximum height it reaches is closest to which of the following ?

  • (a)9.8 metres
  • (b)19.6 metres
  • (c)39.2 metres
  • (d)4.9 metres

  Answer(b) 19.6 metres — the maximum height is u squared divided by twice g, that is 19.6 multiplied by 19.6 and divided by 19.6, which gives 19.6 metres. The time taken to reach that height is u divided by g, which is 2 seconds, and the ball returns to the thrower after a further 2 seconds with the same speed of 19.6 metres per second.