Question

A water hose is used to fill a large cylindrical storage tank of diameter D and height 2D. The hose shoots the water at 45∘ above the horizontal from the same level as the base of the tank and is a distance 6D away. For what range of launch speeds (v0) will the water enter the tank? Ignore air resistance, and express your answer in terms of D and g.

Answer 1

To determine the range of launch speeds for the water to enter the tank, we use the projectile motion formulas and set the constraints that the water must cover a horizontal distance of 6D and not exceed a vertical height of 2D.

Step-by-step explanation:

The question is asking for the range of launch speeds v0 required for water to enter a tank when shot from a hose at a 45° angle. To find this, we can use the equation for projectile motion:

R = (v0^2/g) * sin(2θ), where R is the range, v0 is the initial launch speed, g is the acceleration due to gravity, and θ is the launch angle. We will also use the vertical distance equation: y = v0 * sin(θ) * t - 0.5 * g * t^2.

For the water to enter the tank, it must travel a horizontal distance of 6D to go from the hose to the tank. Since the tank is a distance D in diameter and has a height of 2D, the maximum height the water should achieve is 2D (the height of the tank).

Setting the two equations up with these constraints, solving for v0, and assuming g = 9.8 m/s^2, we can calculate the minimum and maximum launch speeds that will allow the water to enter the tank without overshooting or falling short.