First slide

Friction

Question

An aeroplane of mass M requires a speed v for take off. The length of the runway is s and the coefficient of friction between the tyres and the ground is $μ$ . Assuming that the plane accelerates uniformly during the take-off, the minimum force required by the engine of the plane for take-off is given by

Moderate

A

$M (\frac{v^{2}}{2 s} + μg)$
B

$M (\frac{v^{2}}{2 s} - μg)$
C

$M (\frac{2 v^{2}}{s} + 2 μg)$
D

$M (\frac{2 v^{2}}{s} - 2 μg)$

Solution

The required force is to (i) accelerate the plane from rest to a speed v over a distance s and (ii) to overcome the force of friction ( $= μR = μMg$ ). The acceleration a required to impart a speed v in a distance s is given by $v^{2} - u^{2} = 2 as$ . Since, u = 0, we have $v^{2} = 2 as or a = v^{2} / 2 s$ . The force needed to produce this acceleration is
$F_{1} = mass \times acceleration = \frac{{Mv}^{2}}{2 s}$
The force needed to overcome the force of friction is $F_{2} = μMg$
$∴$ Total force needed $= F_{1} + F_{2} = M (\frac{v^{2}}{2 s} + μg)$

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