Question

a) The speed of an aeroplane is 1200m/s. The engines take in 80 kg of air per second and
mix it with 40 kg of fuel. This mixture is expelled after it ignites and it moves at a
velocity of 3000m/s relative to the aeroplane. Calculate the thrust of the engine.
b) A 2400 W engine pulls a 200 kg block at constant speed up a 12.0 m long, 25.0° incline.
Determine how long does it takes to cover this distance.

Answer 1

Answer on Question #39585, Physics, Mechanics | Kinematics | Dynamics

a) The speed of an aeroplane is $1200\mathrm{m / s}$ . The engines take in $80\mathrm{kg}$ of air per second and mix it with $40\mathrm{kg}$ of fuel. This mixture is expelled after it ignites and it moves at a velocity of $3000\mathrm{m / s}$ relative to the aeroplane. Calculate the thrust of the engine.

b) A 2400 W engine pulls a $200\mathrm{kg}$ block at constant speed up a $12.0\mathrm{m}$ long, $25.0{}^{\circ}$ incline. Determine how long does it takes to cover this distance.

Solution:

a) Thrust is the force which moves an aircraft through the air. Thrust is generated by the propulsion system of the airplane.

Thrust is a mechanical force which is generated through the reaction of accelerating a mass of gas, as explained by Newton's third law of motion. A gas or working fluid is accelerated to the rear and the engine and aircraft are accelerated in the opposite direction.

From Newton's second law of motion, we can define a force $F$ to be the change in momentum of an object with a change in time. Momentum is the object's mass $m$ times the velocity $V$ . So, between two times $t1$ and $t2$ , the force is given by:

$$F = \frac {(m V) _ {2} - (m V) _ {1}}{t _ {2} - t _ {1}}$$ $$F = \Delta p / \Delta t$$

Relative to the aircraft, the initial velocity of the air is $1200\mathrm{m / s}$ (backwards) and the final velocity is $3000\mathrm{m / s}$ (also backwards).

Mass of air per second $m_{1} = 80\mathrm{kg}$

The fuel is carried on the aircraft so it has initial velocity 0 and final velocity $3000 \, \text{m/s}$ .

Mass of fuel per second $m_{2} = 40\mathrm{kg}$

Change of momentum in 1 second $(t_2 - t_1)$

$$\Delta p = m _ {1} \left(v _ {2} - v _ {1}\right) + m _ {2} \left(v _ {2}\right) = 8 0 \cdot (3 0 0 0 - 1 2 0 0) + 4 0 \cdot 3 0 0 0 = 2 6 4 0 0 0 N s$$ $$F = 2 6 4 0 0 0 / 1 = 2 6 4 0 0 0 N$$

b) A 2400 W engine pulls a $200\mathrm{kg}$ block at constant speed up a $12.0\mathrm{m}$ long, $25.0{}^{\circ}$ incline. Determine how long does it takes to cover this distance.

Power $P = 2400\mathrm{W}$

mass $m = 200\mathrm{kg}$

distance $d = 12.0\mathrm{m}$

angle of incline $\theta = 25.0{}^{\circ}$

time $= ?$

Power may be defined as the rate of doing work or the rate of using energy.

Power = Work / time

$$P = \frac {W}{t}$$

Work is equal to potential energy of the block on the height $h$.

$$W = m g h$$ $$h = d \sin \theta$$

Time is

$$t = \frac {W}{P} = \frac {m g d \sin \theta}{P}$$ $$t = \frac {200 \cdot 9.8 \cdot 12.0 \cdot \sin 25{}^{\circ}}{2400} = 4.142 \text{ s}$$

Answer. a) 264000 N,

b) 4.142 s.