# Torricelli’s theorem – problems and solutions

1. A tube with the height of 100 cm filled with water. A hole Q located at 10 cm above the ground. What is the horizontal distance (x)?

__Known :__

Distance between hole and the surface of the water (h) = 100 cm – 10 cm = 90 cm = 0.9 m

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted:__ Distance of x

__Solution :__

**The speed of the water flow at the hole **

*v = **speed**, g = **acceleration due to gravity**, h = **distance between the hole and the surface of the water*

The speed of the water flow at the hole :

**Time in air**

The motion of water from the hole to the ground is the projectile motion. The projectile motion could be understood by analyzing the horizontal and vertical component of the motion separately. The x motion occurs at a constant velocity and the y motion occurs at a constant acceleration of gravity.

In this problem, vertical motion analyzed as free fall motion.

Calculate time in air using the equation of the free fall motion.

__Known :__

The height of hole (y) = 10 cm = 0.1 m

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted :__ Time interval (t)

__Solution :__

y = 1/2 g t^{2}

0.1 = 1/2 (10) t^{2}

0.1 = 5 t^{2}

t^{2 }= 0.1 / 5

t^{2 }= 0.02

t = √0.02 seconds

**The horizontal distance (x) :**

__Known :__

The initial velocity (v_{o} = v_{ox}) = 3√2 m/s

Time in air (t)= √0.02 seconds

__Wanted : __The horizontal distance (x)

__Solution :__

v = x / t

x = v t = (3√2)(√0.02) = (3)(1.41)(0.14) = 0.59 = 0.6 meters

2. A tank containing water with height of 1 meter. At point P, there is a hole. What is the speed of water flow at the hole. Acceleration due to gravity is 10 m/s^{2}.

__Known :__

Distance between hole and the surface of the water (h) = 100 cm – 80 cm = 20 cm = 0.2 m

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted:__ Speed of the water flow at the hole (v)

__Solution :__

The speed of the water flow at the hole :

3. A large tub contains water and there is a faucet as shown in the picture below. If g = 10 ms^{-2}, then the water velocity out of the faucet is…

__Known :__

Height (h) = 85 cm – 40 cm = 45 cm = 0.45 meters

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted:__ Speed of water (v)

__Solution :__

Torricelli’s theorem states that the velocity of water through a hole distant h from the surface of water equals the speed of free falling water from a height of h.

Water velocity is calculated using the free fall motion formula *v*_{t}^{2}* = 2 g h*

v_{t}^{2} = 2 g h = 2(10)(0.45) = 9

v_{t} = √9 = 3 m/s

4. A tub filled with water and on a wall there is a hole (see figure below). The speed of water coming out of the hole is… (g = 10 ms^{-2})

__Known :__

Height (h) = 1.5 m – 0.25 m = 1.25 meters

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted :__ Speed of water (v)

__Solution :__

v_{t}^{2} = 2 g h = 2(10)(1.25) = 25

v_{t} = √25 = 5 m/s

5. A tank containing water as high as 1 meter (g = 10 ms^{-2}) and on the wall there is a leak hole (see figure below). The speed of water coming out of the hole is …

__Known :__

Height (h) = 1 m – 0.20 m = 0.8 meters

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted :__ Speed of water (v)

__Solution :__

v_{t}^{2} = 2 g h = 2(10)(0.8) = 16

v_{t} = √16 = 4 m/s