A tank full of water has a height H = 12:5 m. A small hole is drilled into its side h = 2:5 m above the ground. What is the horizontal distance from the tank to the point where the water spout owing from the hole reaches the ground?and at what height must the hole be drilled in order to
maximize this distance?
To find the horizontal distance from the tank to the point where the water spout reaches the ground, we can use the principles of projectile motion.
First, we need to determine the initial velocity of the water spout as it leaves the hole. We can calculate this velocity using the equation: v = √(2gh), where v is the velocity, g is the acceleration due to gravity, and h is the height of the hole.
Given h = 2.5 m, we can substitute these values into the equation: v = √(2 * 9.8 * 2.5) = √(49) = 7 m/s.
Now, let's find the time it takes for the water spout to reach the ground. We can use the equation: t = √(2 * H / g), where t is the time, H is the height of the water tank, and g is the acceleration due to gravity.
Given H = 12.5 m, we can calculate: t = √(2 * 12.5 / 9.8) = √(25/9.8) = √2.55 ≈ 1.6 s.
Using this time, we can find the horizontal distance covered by the water spout using the equation: d = v * t, where d is the horizontal distance.
Substituting the values, d = 7 * 1.6 = 11.2 m.
Therefore, the horizontal distance from the tank to the point where the water spout reaches the ground is approximately 11.2 meters.
To determine the height at which the hole must be drilled in order to maximize this distance, we need to consider the concept of projectile motion. In projectile motion, the horizontal distance covered is maximized when the angle of projection is 45 degrees.
Since the water is flowing horizontally from the hole, we can conclude that the angle of projection is 45 degrees when the water spout will reach the greatest horizontal distance. Thus, we can say that the height at which the hole must be drilled to maximize the horizontal distance is when it is at half the height of the water tank.
In this case, since the height of the water tank is H = 12.5 m, the optimal height at which the hole should be drilled is h = H/2 = 12.5/2 = 6.25 m.
Therefore, to maximize the horizontal distance, the hole must be drilled at a height of 6.25 meters.