A stream of water flows through
a small hole at depth h = 10 cm
in a tank holding water to height
H =40 cm. At what distance x (in
cm) does the stream strike the
floor? (Neglect air resistance).
To find the distance at which the stream of water strikes the floor, we can use the principle of conservation of energy.
When the water flows out of the hole, it possesses both kinetic energy and potential energy. The potential energy is converted to kinetic energy as the water falls.
The potential energy of water at depth h can be calculated using the formula:
Potential Energy = m * g * h
where m is the mass of water and g is the acceleration due to gravity.
Since the depth h is given as 10 cm, we can assume a small cross-sectional area for the hole and consider the velocity of the water stream as constant. Therefore, the mass of water flowing out per second would also be constant.
The equation for conservation of energy can be written as:
Potential Energy = Kinetic Energy
m * g * h = ½ * m * v^2
where v is the velocity of the water stream.
Simplifying the equation:
g * h = ½ * v^2
Solving for velocity v:
v = sqrt(2 * g * h)
Substituting the values of g (acceleration due to gravity) as 9.8 m/s^2, and h as 10 cm (0.1 m), we can calculate the velocity v.
v = sqrt(2 * 9.8 * 0.1) ≈ 3.13 m/s
Now, to find the horizontal distance x at which the stream of water strikes the floor, we can use the equation of motion for horizontal motion:
Distance = velocity * time
At the instant when the water strikes the floor, the time taken would be the same for both horizontal and vertical motion. We can find this time using the vertical motion equation:
H = ½ * g * t^2
where H is the height of the water in the tank.
Simplifying the equation:
t = sqrt(2 * H / g)
Substituting the values of H (40 cm) and g (9.8 m/s^2), we can calculate the time t.
t = sqrt(2 * 0.4 / 9.8) ≈ 0.28 s
Now, using this time, we can calculate the horizontal distance x using the equation of motion:
x = v * t
Substituting the values of v (3.13 m/s) and t (0.28 s), we can calculate the horizontal distance x.
x = 3.13 * 0.28 ≈ 0.876 m (87.6 cm)
Therefore, the stream of water strikes the floor at a horizontal distance of approximately 87.6 cm from the base of the tank.