The cylindrical vessel is kept on stand. Calculate maximum possible horizontal range of water stream and height (x) of small hole from bottom of vessel respectively :-

To determine the maximum possible horizontal range of a water stream from a hole in a cylindrical vessel and the height of that hole from the bottom, we can use principles from fluid dynamics and projectile motion. Let's break this down step by step.

Understanding the Problem

When water flows out of a hole in a vessel, it behaves like a projectile. The horizontal range and the height of the hole are influenced by the speed of the water exiting the hole and the height of the water column above it. The key factors to consider are:

• The height of the hole from the bottom of the vessel (h).
• The gravitational acceleration (g), which is approximately 9.81 m/s².
• The velocity of the water as it exits the hole, which can be derived from Torricelli's theorem.

Applying Torricelli's Theorem

According to Torricelli's theorem, the speed (v) of fluid flowing out of an orifice under the influence of gravity is given by:

v = √(2gh)

Here, h is the height of the water column above the hole. This speed is crucial for calculating the horizontal range.

Calculating the Horizontal Range

Once the water exits the hole, it follows a projectile motion. The horizontal range (R) can be calculated using the formula:

R = v * t

Where t is the time the water takes to hit the ground. To find t, we can use the formula for free fall:

t = √(2h/g)

Substituting the expression for v into the range formula gives us:

R = √(2gh) * √(2h/g)

This simplifies to:

R = 2h

Finding the Height of the Hole

To maximize the horizontal range, the height of the hole should ideally be at the maximum height of the water column. Therefore, if we want to find the height (x) of the hole from the bottom of the vessel, we can set:

x = h

Thus, the height of the hole from the bottom of the vessel is equal to the height of the water column above it.

Final Results

In summary, the maximum possible horizontal range of the water stream is:

R = 2h

And the height (x) of the small hole from the bottom of the vessel is:

x = h

By understanding these principles, you can see how fluid dynamics and projectile motion work together to determine the behavior of water flowing from a vessel. If you have any further questions or need clarification on any part of this process, feel free to ask!