Physics, 30.08.2019 19:50 xojade

An airplane is heading due south at a speed of 640 km/h. if a wind begins blowing from the southwest at a speed of 70 km/h (average), calculate the following.
(a) the velocity (magnitude and direction) of the plane relative to the ground
643.81
km/h
° (east of south)
(b) how far from its intended position will it be after 10 min if the pilot takes no corrective action. [hint: first draw a diagram.]

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Answer from: tavito968

(a) 609∠273°

(b) 5.31 km far away from the desired point to the east

5.31 far away from the desired point to the north

Explanation:

(a) In polar coordinates, the speed of the airplane is the next vector 640∠270° (640 km/h is the modulus and 'heading due south' is equivalent to an angle of 270° between the speed vector and the positive x-axis). On its part, the speed of the wind is the next vector 70∠45°.

The vector 640∠270° can be decomposed into Vx (x-coordinate component) and Vy (y-coordinate component) as follows:

Vx1 = 640*cos(270°) = 0

Vy1 = 640*sin(270°) = -640

For 70∠45°:

Vx2 = 70*cos(45°) = 31.82

Vy2 =70*sin(45°) = 31.82

The velocity of the airplane (V3) is calculated as follows:

Vx3 = Vx1 + Vx2 = 0 + 31.82 = 31.82

Vy3 = Vy1 + Vy2 = -640 + 31.82 = -608.18

The magnitude of the speed is:

V3 = √(Vx3² + Vy3²)

V3 = √(31.82² + (-608.18)²) = 609 km/h

α = tan (Vy3/Vx3)

α = tan (-608/31.82) = 273° = -87°

(b) Given that:

speed = displacement/time

Then:

displacement = speed *time

At 640 km/h and after 10 min (= 1/6 h)

displacement = 640 * 1/6 = 106.67 km

This is the desired displacement, that is, in south direction.

At 609 km/h and after 10 min (= 1/6 h)

displacement = 609 * 1/6 = 101.5 km

Real displacement, in east of south direction (forming an angle of -87° with the positive x-coordinate) . This point is at the following distance form the desired point:

cos 87° * 101.5 = 5.31 km far away from the desired point to the east

106.67 - sin 87° * 101.5 = 5.31 far away from the desired point to the north

Answer from: Quest

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Answer from: Quest

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

$x(t)=ax(x-6)$

where $a$ is unknown. because the position is parabolic, there is symmetry about the midpoint at $t=3$ , and we know that

$x(3)=10\implies -9a=10\implies a=-\dfrac{10}9$

so that the position function is

$x(t)=-\dfrac{10}9x(x-6)$

which means the velocity is

$v(t)=\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac{20}9x+\dfrac{60}9$

then the total distance traveled is

$\displaystyle\int_{t=0}^{t=6}|v(t)|\,\mathrm dt=\left\{\int_{t-0}^{t=3}-\int_{t=3}^{t=6}\right\}v(t)\,\mathrm dt=20$

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at $x=0$ and ends at $x=0$ , so its displacement is 0.

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An airplane is heading due south at a speed of 640 km/h. if a wind begins blowing from the southwest...

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