Maximum height for water from a fire hose

[tuki]

Homework Statement

Fire hose has diameter of 4.0 cm and flow rate of 10 L/s. There is pressure of 2.2 bar inside the hose. How high the water can go at best? Water density is 1.00E3 kg/m^3 and air pressure outside the hose is 1.0 bar.

Homework Equations

Flow rate
$$ Q = Av $$
Newtons equations?

The Attempt at a Solution

I tired to find speed for water particle when it exits the hose. I can get speed directly from flow rate.

$$ v = \frac{Q}{A} = \frac{Q}{(\frac{d}{2})^2 \pi} $$
$$ v \approx 7.957 \text{ m/s} $$

If we would aim the hose at 45 angle it would follow height as function of time as:

$$ h(t) = vt-gt^2 $$

i solved the maxima with first derivative $$ t = \frac{v}{2g} \approx 0.4055 \text{ s} $$

then you would compute the height from the function $$ h(0.41\text{ s}) \approx 1.6 \text{ m} $$

However the correct answer should be 26m. What I'am doing wrong?

 

[kuruman]

Try Bernoulli's equation and don't forget the necessary unit conversions.

 

[BvU]

hoses have nozzles to convert pressure to speed
kuru was faster to point you in the right direction
note they ask for maximum height, so I d aim a little higher

 

[tuki]

hoses have nozzles to convert pressure to speed
kuru was faster to point you in the right direction
note they ask for maximum height, so I d aim a little higher

Try Bernoulli's equation and don't forget the necessary unit conversions.

I tried Bernoulli's equation and ended up with this.
$$ p_1 + \frac{1}{2}pv^2_1 = p_2 + \frac{1}{2}pv_2^2 $$
Speed can be expressed as
$$ v = \frac{Q}{\pi (\frac{d}{2})^2} $$
$$ \implies v_2 = \sqrt{\frac{p_1-p_2}{\rho}+(\frac{Q}{(\frac{d}{2})}^2 \pi)^2\cdot 2} $$
when i compute with numbers i get 19m/s for speed.

Computing maximum height with mgh = 1/2 mv² we get that h = v²/2g, which gives approx 18.3 m.
Correct answer was 26 m. Any suggestion on what I'am doing wrong?

 

[kuruman]

Since the problem is asking for the speed, use Bernoulli's equation between a point just inside the hose and the point at maximum height. You don't really need the speed. Also, I get 26 m if I assume that the pressure outside is zero. It looks like there is an error in the quoted answer that needs to be checked.