A ball thrown vertically upwards from level ground is...

A ball thrown vertically upwards from level ground is observed twice, at a height H above the ground, within a time interval t. The initial velocity of the ball was (a) $\sqrt{8 \mathrm{gH}+\mathrm{g}^{2} \mathrm{t}^{2}}$ (b) $\sqrt{8 \mathrm{gH}+\left(\frac{\mathrm{gt}}{2}\right)^{2}}$ (c) $\frac{1}{2} \sqrt{8 \mathrm{gH}+\mathrm{g}^{2} \mathrm{t}^{2}}$ (d) $\sqrt{8 \mathrm{gH}+4 \mathrm{~g}^{2} \mathrm{t}^{2}}$

A ball thrown vertically upwards from level ground is observed twice, at a height H above the ground, within a time interval t. The initial velocity of the ball was
(a) $\sqrt{8 \mathrm{gH}+\mathrm{g}^{2} \mathrm{t}^{2}}$
(b) $\sqrt{8 \mathrm{gH}+\left(\frac{\mathrm{gt}}{2}\right)^{2}}$
(c) $\frac{1}{2} \sqrt{8 \mathrm{gH}+\mathrm{g}^{2} \mathrm{t}^{2}}$
(d) $\sqrt{8 \mathrm{gH}+4 \mathrm{~g}^{2} \mathrm{t}^{2}}$

Key Concepts

Projectile Motion Symmetry

In projectile motion under gravity, when an object reaches a certain height, it does so twice: once while ascending and once while descending. The symmetry of these two events about the apex of the trajectory allows for relationships between the times at which these events occur. This symmetry is crucial for deriving results about the object's initial velocity or altitude when given a specific time interval between the two occurrences at the same height.

Quadratic Kinematic Equations

The kinematic equation for displacement in uniformly accelerated motion is quadratic in time, leading to two solutions when solving for the time at which a particular height is reached. This quadratic nature is important in understanding that an object can pass the same height at two different moments, and it enables the use of algebraic methods—such as summing the roots—to find relationships between the motion parameters, including the initial velocity.

Uniformly Accelerated Motion

This concept involves the motion of objects under a constant acceleration, such as gravity. The key equations relate displacement, initial velocity, time, and acceleration, and are foundational for analyzing projectile motion. By using these equations, one can predict the position and velocity of an object at any time, which is essential for solving problems involving objects thrown vertically upward or downward.

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