A tank of water 3 meters deep sits on a table 1 meter tall. A hole is located 2 meters from the bottom of the tank. Water flows out of the hole and strikes the ground. Where does it strike the ground? Assume incompressible, non viscous flow.
To determine where the water strikes the ground, we can analyze the motion of the water as it flows out of the hole.
The water molecules leaving the hole will follow a curved path due to the force of gravity acting on them. This path can be approximated as a parabolic trajectory.
To find where the water strikes the ground, we need to determine the horizontal distance traveled by the water from the hole to the ground. We can break this down into two parts:
1. Horizontal distance traveled in the vertical direction:
- This is the displacement of the water in the vertical direction due to the initial height difference between the hole and the ground.
- Given that the tank is 3 meters deep and the table is 1 meter tall, the initial vertical displacement is 3 meters - 1 meter = 2 meters.
2. Horizontal distance traveled horizontally:
- This is the displacement of the water in the horizontal direction due to its motion.
- The water will follow a parabolic trajectory, and the horizontal distance traveled can be determined using projectile motion equations.
The key variables we need to consider are:
- Initial vertical displacement (h₀) = 2 meters
- Initial velocity in the vertical direction (v₀) = 0 m/s (the water is not initially moving)
- Acceleration due to gravity (g) = 9.8 m/s² (assuming Earth's gravity)
Using the equation of motion in the vertical direction, h = h₀ + v₀t + (1/2)gt², we can solve for the time it takes for the water to reach the ground:
0 = 2 m + 0 m/s * t + (1/2)(9.8 m/s²)t²
Simplifying this equation, we get:
4.9t² = 2
Solving for t, we find t ≈ 0.645 s (rounded to three decimal places).
Now, using the time it takes for the water to reach the ground, we can calculate the horizontal displacement using the equation x = vt, where v represents the horizontal velocity.
Since water flows out of the hole due to gravity, the horizontal velocity (v) remains constant throughout the trajectory. Therefore, the horizontal distance traveled by the water is given by:
x = v * t
To find the horizontal velocity, we can use the fact that the horizontal velocity is constant in the absence of external forces (incompressible and non-viscous flow). Therefore, the horizontal velocity can be determined by considering a horizontal line from the bottom of the tank to the hole. The velocity of water at the hole will be equal to the velocity of water at the bottom of the tank.
The velocity of water at the bottom of the tank can be determined using the equation v = √(2gh), where h represents the depth of the tank:
v = √(2 * 9.8 m/s² * 3 m) = √(58.8) ≈ 7.67 m/s (rounded to two decimal places)
Now, using the calculated horizontal velocity and the time it takes for the water to reach the ground, we can find the horizontal distance traveled by the water:
x = v * t = 7.67 m/s * 0.645 s ≈ 4.95 m (rounded to two decimal places)
Therefore, the water strikes the ground approximately 4.95 meters from the base of the tank (or 2 meters from the hole).