An engine pumps water through a hose pipe.Water passes through the pipe and leaves it with a velocity of 2 m/s. The mass per unit length of water in the pipe is 100 kg/m. What is the power of the engine? [2010]a)400 Wb)200 Wc)100 Wd)800 WCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

We know, power = Force x velocity

P = Fv (1)

we also know Force = mass x acceleration

or, Force = mass x change in velocity/time taken

if intial velocity = 0,

then, Force = mass x velocity/time taken

F = m(v/t) (2)

put equation (2) in equation (1),

P = m(v/t)v = mv^2 / t

now, mass = density x volume

or , mass = density x base area x length

so, P = v^2.p.A.1/ t

P = (pA)v^3

here, pA

is mass per unit length . e.g.,

= 100kg/m

hence, P = 100 x (2)^3 = 100 x 8 = 800W

So, option D is correct answer.

Answer

We know, power = Force × velocity
e.g., P = Fv ---->(1)

we also know Force = mass × acceleration
or, Force = mass × change in velocity/time taken
if intial velocity = 0,
then, Force = mass × velocity/time taken
e.g., F = m(v/t) ---->(2)

put equation (2) in equation (1),
P = m(v/t)v = mv²/t
now, mass = density × volume
or , mass = density × base area × length
e.g., m =density*A*l , put it in above equation.

so, P = v^2*(density) *A^(l/t)
P = ((density) *A) v^³

here,(density) *A is nothing mass per unit length . e.g., n= 100kg/m

hence, P = 100 × (2)³ = 100 × 8 = 800W

Answer

Given data:
- Velocity of water leaving the pipe (v) = 2 m/s
- Mass per unit length of water in the pipe (μ) = 100 kg/m

We need to find the power of the engine.

The power (P) can be calculated using the formula:
P = Force × Velocity

But before calculating the force, we need to determine the mass flow rate (ṁ) of water through the pipe.

Mass flow rate is given by the formula:
ṁ = μ × A × v

- A is the cross-sectional area of the pipe.

Since the mass per unit length of water is given, we can calculate the cross-sectional area using the formula:
A = μ⁻¹

Therefore, the mass flow rate becomes:
ṁ = v/μ

Now, let's calculate the mass flow rate:
ṁ = 2/100 = 0.02 kg/s

The force (F) exerted by the water on the engine can be calculated using Newton's second law of motion:
F = ṁ × Δv

- Δv is the change in velocity of the water.

As the water leaves the pipe with a velocity of 2 m/s, Δv = 2 m/s

Substituting the values, we get:
F = 0.02 × 2 = 0.04 N

Now, let's calculate the power:
P = F × v = 0.04 × 2 = 0.08 W

But the answer options are given in watts (W), so we need to convert the power to watts:
1 W = 1 J/s
0.08 W = 0.08 J/s = 80 W

The correct answer is option D) 800 W.

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