A stone is dropped frompoint O . Then are points P, Q, R along its paths,If OP, OQ,OR.. are in GP then prove that velocities are also in GP

To tackle this problem, we need to analyze the motion of the stone dropped from point O and how the distances OP, OQ, and OR relate to the velocities at those points. The key here is to understand the relationship between distance and velocity in uniformly accelerated motion, specifically under the influence of gravity.

Understanding the Motion of the Stone

When a stone is dropped from a height, it accelerates downwards due to gravity. The distance fallen after a certain time can be described by the equation:

s = ut + (1/2)gt²

In this case, the initial velocity (u) is zero because the stone is dropped, so the equation simplifies to:

s = (1/2)gt²

Points in Geometric Progression

Let’s denote the distances OP, OQ, and OR as s1, s2, and s3 respectively. If these distances are in geometric progression (GP), we can express this relationship mathematically:

• s2² = s1 * s3

This means that the middle term squared is equal to the product of the other two terms. Now, since we know the distances are related to time squared (as shown in the distance equation), we can express these distances in terms of time:

• s1 = (1/2)g(t1)²
• s2 = (1/2)g(t2)²
• s3 = (1/2)g(t3)²

Finding Velocities

The velocity of the stone at any point can be calculated using the formula:

v = u + gt

Since the initial velocity (u) is zero, the velocity at each point becomes:

• v1 = gt1
• v2 = gt2
• v3 = gt3

Proving Velocities are in GP

Now, we need to show that v1, v2, and v3 are also in GP. We can express the velocities in terms of the distances:

Using the distances in GP, we have:

• v1 = g√(2s1/g) = √(2gs1)
• v2 = g√(2s2/g) = √(2gs2)
• v3 = g√(2s3/g) = √(2gs3)

Now, we can check if these velocities are in GP:

For v2² = v1 * v3, we substitute:

v2² = 2gs2

v1 * v3 = √(2gs1) * √(2gs3) = 2g√(s1 * s3)

Since s1, s2, and s3 are in GP, we know that:

s2² = s1 * s3

Thus, we can conclude:

v2² = v1 * v3

Final Thoughts

This shows that the velocities v1, v2, and v3 are indeed in geometric progression. The relationship between the distances and the velocities in uniformly accelerated motion under gravity leads us to this conclusion. So, if the distances OP, OQ, and OR are in GP, the corresponding velocities at those points will also be in GP.