Francesca, who likes physics experiments, dangles her watch from a thin piece of string while the jetliner she is in takes off from Dulles Airport. She notices that the string makes an angle of 25° with respect to the vertical as the aircraft accelerates for takeoff, which takes about 18s. Estimate the takeoff speed of the aircraft.

When a suspended string makes and angle of 25 degree while aircraft is accelerating on runway then we can use force equations to find the acceleration of the aircraft

So we will have

[tex]T sin\theta = ma[/tex]

[tex]T cos\theta = mg[/tex]

now from above two equations we have

[tex]a = g tan\theta[/tex]

[tex]a = (9.81) tan25[/tex]

[tex]a = 4.57 m/s^2[/tex]

now if it took 18 s to take off we can have final speed of the aircraft given by

[tex]v_f - v_i = at[/tex]

[tex]v_f - 0 = (4.57)(18)[/tex]

[tex]v_f = 82.3 m/s[/tex]

AFOKE88 AFOKE88 · Answer 2 · 2021-11-07T10:13:07+01:00

The takeoff speed of the aircraft is gotten as;

v = 82.26 m/s

We are given;

Angle made by the string with respect to the vertical; θ = 25°

Time; t = 18 s

Now, this motion here is similar to that of an object placed on a very smooth wedge resting on an inclined surface at an angle θ without friction. Now, when the mass is released, the acceleration will be expressed as;

a = g tan θ

In this case in question, after the aircraft accelerates for the takeoff, the acceleration will be;

a = g tan 25°

g is acceleration due to gravity has a constant value of 9.8 m/s²

Thus;

a = 9.8 × 0.4663

a = 4.57 m/s²

Now, using newton's first equation of motion we can find the takeoff speed from;

v = u + at

We are told it takes 18 seconds to take off and as a result initial velocity is zero and the take off speed will be its' final speed.

Thus;

v = 0 + (4.57 × 18)

v = 82.26 m/s

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