During a training session, two footballers pass a ball repeatedly between each other. The 'to and fro' motion…

Analyze the question: We are asked why the to-and-fro passing of a ball between two footballers is not simple harmonic motion (SHM). We must decide which of the three statements correctly give reasons.

Concept: Simple harmonic motion is motion in which:

• the acceleration is always directed towards a fixed point (the equilibrium or centre position), and
• the acceleration is directly proportional to the displacement from that point, and opposite to it: a = −ω²x.

Because force = mass × acceleration, a restoring force directed towards the centre (F = −kx) is also required.

Check each statement:

• I. Acceleration of the ball is not directed towards a fixed point. This is true. When the ball is in flight, its only significant acceleration is due to gravity (downwards), plus a sudden change when it is kicked. It does not point towards a fixed centre point between the players.
• II. Restoring force is not directed towards the centre. This is true. There is no restoring force pulling the ball back towards a midpoint. The force comes from the kicks, which push the ball in whichever direction the player chooses, and from gravity, which acts downwards.
• III. Acceleration is not directly proportional to displacement from a fixed point. This is true. The ball's acceleration is roughly constant (g) in flight and does not vary with displacement from any fixed point, so a ∝ −x does not hold.

All three statements correctly explain why the motion is not SHM.

Answer: I, II and III, which is option D.

Tip: For "why is this not SHM?" questions, recall the three SHM conditions: (1) acceleration directed towards a fixed point, (2) a restoring force towards the centre, and (3) acceleration proportional to displacement. If a motion fails them, all the matching statements are valid reasons. Keep going, you're doing well!