Physics 9702 · AS & A Level · Momentum and Newton's laws of motion

Momentum and Newton's laws of motion — practice question

A small remote-controlled model aircraft has two propellers, and each one has a diameter of $16\,\text{cm}$. Fig. 3.1 shows the aircraft from the side while it is hovering. Each propeller drives air vertically downward so that the aircraft stays at a fixed position. The density of the air is $1.2\,\text{kg m}^{-3}$. Assume that the air leaving each propeller travels at a steady speed of $7.6\,\text{m s}^{-1}$ through a uniform cylinder of diameter $16\,\text{cm}$. Also assume that the air above each propeller is stationary.
(a)[3]

Show that, over a time interval of $3.0\,\text{s}$, one propeller drives down a mass of air of $0.55\,\text{kg}$.

(b(i))[1]

Calculate the increase in momentum of the mass of air from (a).

(b(ii))[1]

Calculate the downward force that the propeller exerts on this mass of air.

(c(i))[1]

State the upward force on one propeller.

(c(ii))[1]

State the name of the law that describes the connection between the force in (b)(ii) and the force in (c)(i).

(d)[1]

Determine the aircraft’s mass.

(e)[1]

In order for the aircraft to hover at a much greater altitude (height), the propellers must send the air downward at a greater speed than when the aircraft hovers at a low altitude. Suggest the reason for this.

(f)[2]

While the aircraft is hovering high above the ground, an electrical fault makes the propellers stop turning. The aircraft then falls vertically. Once it is moving at the constant speed of $22\,\text{m s}^{-1}$, an alarm on board emits sound of frequency $3.0\,\text{kHz}$. The speed of sound in the air is $340\,\text{m s}^{-1}$. Determine the frequency heard by a person standing directly below the falling aircraft.

Worked solution & mark scheme

This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: Apply $\rho = \dfrac{m}{V}$

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