(i) A particle thrown vertically up so as to reach a maximum...

(i) A particle thrown vertically up so as to reach a maximum height $H$ in time $T$. It is at height $h$ in time $t$. Show that $\frac{t}{T}=1 \pm \sqrt{1-\frac{h}{H}}$ (ii) Particle A is thrown vertically up and after some time particle B is thrown vertically up from the same point with the same velocity. They collide 7 seconds after particle A was thrown and at a height $7 / 16$ of maximum height reached by A. What was the time interval between their projections?

(i) A particle thrown vertically up so as to reach a maximum height $H$ in time $T$. It is at height $h$ in time $t$. Show that $\frac{t}{T}=1 \pm \sqrt{1-\frac{h}{H}}$
(ii) Particle A is thrown vertically up and after some time particle B is thrown vertically up from the same point with the same velocity. They collide 7 seconds after particle A was thrown and at a height $7 / 16$ of maximum height reached by A. What was the time interval between their projections?

Key Concepts

Quadratic Equations in Kinematics

When applying the kinematic equations to problems involving projectile motion, one often encounters quadratic equations in terms of time. Solving these quadratic equations provides the possible time values at which the object reaches a particular height. Understanding the properties of quadratic equations is thus pivotal in analyzing motion under constant acceleration.

Projectile Motion Under Gravity

This concept deals with the motion of an object that is thrown or projected into the air and is subject only to gravitational acceleration. The motion is effectively one-dimensional in the vertical direction, and its analysis involves understanding how gravity influences the object's velocity and displacement over time.

Maximum Height and Time of Flight

The maximum height in projectile motion is the highest vertical position reached by the object, which occurs when its upward velocity becomes zero. The time to reach maximum height and the total time of flight are directly connected to the initial velocity and the acceleration due to gravity. These concepts are crucial for analyzing the vertical component of projectile motion.

Kinematic Equations for Uniform Acceleration

Kinematic equations describe the relationships between displacement, velocity, acceleration, and time for objects moving with constant acceleration. These equations are essential for solving problems involving projectile motion, as they allow one to determine key variables like position or time at any point during the motion.

Collision Analysis in Vertical Projectile Motion

Collision analysis involves determining the conditions under which two objects moving along the same line (or trajectory) with different starting times or initial velocities meet at the same point at the same time. For vertically thrown objects, this analysis relies on setting the displacement equations equal to each other while accounting for differences in launch times, and it often leads to solving for the unknown time interval between releases.

Relative Motion Concepts

Relative motion in the context of projectile problems involves analyzing the motion of one object with respect to another. This technique is useful when two objects are moving along the same trajectory but with different initial conditions. By examining the differences in their displacements over time, one can determine parameters such as the time interval between their projections or the point of collision.

Please give Ace some feedback