when stones are dropped from top of tower a half a second apart.the time after dropping the first stone at which the distance between two stones is 20m is

To solve the problem of finding the time after dropping the first stone at which the distance between the two stones is 20 meters, we can use the equations of motion under gravity. Let’s break it down step by step.

Understanding the Motion of the Stones

When an object is dropped from a height, it accelerates downwards due to gravity. The acceleration due to gravity (g) is approximately 9.81 m/s². In this scenario, we have two stones: the first stone is dropped at time t = 0 seconds, and the second stone is dropped half a second later, at t = 0.5 seconds.

Position of Each Stone

We can calculate the position of each stone using the formula for distance traveled under constant acceleration:

• Distance (s) = ut + (1/2)gt²

Here, u is the initial velocity (which is 0 since the stones are dropped), g is the acceleration due to gravity, and t is the time in seconds.

Calculating the Distance of Each Stone

For the first stone, which is dropped at t = 0 seconds:

• Distance of first stone (s₁) after time t seconds: s₁ = 0 + (1/2) * 9.81 * t² = 4.905t²

For the second stone, which is dropped at t = 0.5 seconds, the time it has been falling when the first stone has been falling for t seconds is (t - 0.5) seconds:

• Distance of second stone (s₂) after time t seconds: s₂ = 0 + (1/2) * 9.81 * (t - 0.5)² = 4.905(t - 0.5)²

Finding the Distance Between the Two Stones

The distance between the two stones at any time t is given by:

• Distance between stones = s₁ - s₂

We want this distance to equal 20 meters:

• 4.905t² - 4.905(t - 0.5)² = 20

Simplifying the Equation

Now, let’s simplify the equation:

• Expanding (t - 0.5)² gives us: (t - 0.5)² = t² - t + 0.25
• Substituting this back into the equation: 4.905t² - 4.905(t² - t + 0.25) = 20
• This simplifies to: 4.905t² - 4.905t² + 4.905t - 1.22625 = 20
• Thus, we have: 4.905t - 1.22625 = 20

Solving for t

Now, let’s isolate t:

• 4.905t = 20 + 1.22625
• 4.905t = 21.22625
• t = 21.22625 / 4.905 ≈ 4.33 seconds

Final Result

Therefore, the time after dropping the first stone at which the distance between the two stones is 20 meters is approximately 4.33 seconds. This means that about 4.33 seconds after the first stone is dropped, the second stone will be 20 meters away from the first stone. This approach illustrates the principles of kinematics and how objects behave under the influence of gravity.