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3 years ago

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A small plane flies 40.0 km in a direction 60° north of east and then flies 30.0 km in a direction 15° north of east. Use the an

alytical method to find the total distance the plane covers from the starting point, and the geographic direction of its displacement vector. What is its displacement vector?

1 answer:

lana66690 [7]3 years ago

8 0

Answer:

d= 64.7 km

$\theta = 40.9^{o}$

displacement vector $=r_xi + r_yj = 48.9i + 42.4j$

Explanation:

total distance = 40 + 30 = 70 km

during 1st flight

$r_1 x = 40*cos60$

$r_1 x = 20 km$

$r_1 y = 40*sin60$

$r_1 y = 34.64 km$

during 2nd flight

$r_2 x = 30*cos15$

$r_2 x = 28.9 km$

$r_2 y = 30*sin15$

$r_2 y = 7.76 km$

the two component of r are:

$r_x = r_1x + r_2x = 20 + 28.9 = 48.9 km$

$r_y = r_1y + r_2y = 34.64 + 7.76 = 42.4 km$

Geographical Direction $\theta = tan^{-1}\frac{r_y}{r_x} [tex]\theta = 40.9^{o}$

Displacement d $= \sqrt{r_x^2 + r_y^2}$

$d = \sqrt{48.9^2+42.4^2} = 64.7 km$

d= 64.7 km

displacement vector $=r_xi + r_yj = 48.9i + 42.4j$

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What is the instantaneous speed at 5 seconds? (remember s=d/t)

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5. An acrobat, starting from rest, swings freely on a trapeze of

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The energy conservation and trigonometry we can find the results for the questions about the movement of the acrobat are;

a) The maximum speed is v = 4.89 m / s

b) The maximum height is h = 1.22 m

The energy conservation is one of the most fundamental principles of physics, stable that if there are no friction forces the mechanistic energy remains constant. Mechanical energy is the sum of the kinetic energy plus the potential energies.

Em = K + U

Let's write the energy in two points.

Starting point. Highest part of the oscillation

Em₀ = U = m g h

Final point. Lower part of the movement

$Em_f$ = K = ½ m v²

Energy is conserved.

Emo = $Em_f$

m g h = ½ m v²

v² = 2 gh

Let's use trigonometry to find the height, see attached.

h = L - L cos θ

h = L (1- cos θ)

They indicate that the initial angle is tea = 48º and the length is L = 3.7 m, let's calculate.

h = 3.7 (1- cos 48)

h = 1.22 m

this is the maximum height of the movement.

Let's calculate the velocity.

$v= \sqrt{2 \ 9.8 \ 1.22}$

v = 4.89 m / s

In conclusion using the conservation of energy and trigonometry we can find the results for the questions about the movement of the acrobat are;

a) The maximum speed is v = 4.89 m / s

b) The maximum height is h = 1.22 m

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An automobile travels on a straight road for 42 km at 45 km/h. it then continues in the same direction for another 42 km at 90 k

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Answer:

a) The average velocity is v = (60 km/h ; 0)

b) The average speed is 60 km/h

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The velocity is a vector that has a magnitude and direction. The average speed is the distance traveled over time without taking into account the direction of the motion.

a)The average velocity is calculated as the displacement over time:

v = Δx/Δt

where

v = velocity

Δx = final position - initial position = traveled distance relative to the center of the reference system.

Δt = final time - initial time (initial time is usually = 0)

We know that the displacement is 84 km but we do not know the time. It can be calculated from the two parts of the trip.

In part 1:

v = 45 km/h = 42 km / t

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In part 2:

v = 90 km/h = 42 km / t

t = 42 km / 90 km/h

t = 0.47 h

The time of travel is 0.47 h + 0.93 h = 1.4 h

The average velocity will be:

v = 84 km / 1.4 h = 60 km/h

Expressed as a vector in a 2-dimension plane:

v = (60 km/h; 0)

b) The average speed is calculated as the distance traveled over time. Note that in this case, the distance is equal to the displacement since the direction of the motion is always in one direction. But if the direction of the second part of the trip would have been the opposite to the direction of the first part, the displacement would have been 0 (final position - initial position = 0, because final position = initial position), then, the average velocity would have been 0. In change, the average speed would have been the distance traveled (84 km, 42 km in one direction and 42 km in the other) over time.

Then:

average speed = 84 km / 1.4 h = 60 km/h

c) see attached figure.

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