Proof that the particle will be at three fourth of its greatest height at times which are in the ratio 1:3

Understanding the motion of the particle:
- When a particle is projected vertically upwards, it follows a parabolic path due to the gravitational force acting on it.
- The particle reaches its greatest height when its velocity becomes zero at the highest point of its trajectory.

Time taken to reach the greatest height:
- The time taken for the particle to reach its greatest height can be calculated using the equation: time taken = initial velocity / acceleration due to gravity.
- Let the time taken to reach the greatest height be T.

Understanding the concept:
- At any time t after projection, the particle will be at a height h given by the equation: h = initial velocity * t - (1/2) * acceleration due to gravity * t^2.

Proving the given statement:
- Let the times be in the ratio 1:3, i.e., the first time is t and the second time is 3t.
- At time t, the particle will be at a height of h1 = initial velocity * t - (1/2) * acceleration due to gravity * t^2.
- At time 3t, the particle will be at a height of h2 = initial velocity * 3t - (1/2) * acceleration due to gravity * (3t)^2.
- Simplifying h2, we get h2 = 3 * (initial velocity * t - (1/2) * acceleration due to gravity * t^2).
- Since h2 = 3h1, the particle will be at three fourth of its greatest height at times which are in the ratio 1:3.
Therefore, it is proved that the particle will be at three fourth of its greatest height at times which are in the ratio 1:3.