Answer :
Final answer:
The magnitude of the change in velocity is 177.4 km/h, the change in direction of velocity is 45°, and the magnitude of the average acceleration during the trip is 59.1 km/h².
Explanation:
To find the magnitude of the change in velocity, we need to calculate the difference between the initial and final velocities. The initial velocity north is 192 km/h and the final velocity northwest is 250 km/h, making an angle of 45° west of north. Using the Pythagorean theorem, we can find the magnitude of the change in velocity:
Magnitude of change in velocity: √((250 km/h)^2 + (-192 km/h)^2) = 177.4 km/h
To find the change in direction of the velocity, we can use trigonometry. The initial velocity is due north, so it has a direction of 0°. The final velocity is 45° west of north. The change in direction is the difference between these angles:
Change in direction of velocity: 45° - 0° = 45°
The magnitude of the average acceleration during the trip can be calculated using the formula: average acceleration = (change in velocity) / (time taken). As the time taken is 3.0 hours, we can calculate the average acceleration:
Magnitude of average acceleration during trip: (177.4 km/h) / (3.0 h) = 59.1 km/h²
The magnitude of the change in velocity is determined as 408.94 km/h.
The change in direction of the velocity is 64.4⁰ north of west.
The magnitude of the average acceleration during the trip is 0.0105 m/s².
Magnitude of change in velocity
The magnitude of change in velocity is the resultant velocity of the plane.
v² = a² + b² - 2ab cosθ
where;
- θ is the angle between the two velocities = 45 + 90 = 135
v² = (192²) + (250²) - 2(192 x 250) cos(135)
v² = 167,236
v = √167,236
v = 408.94 km/h
Vertical component of the velocity
vyi = 192 km/h
vy2 = 250 x sin(45) = 176.77 km/h
vy(total) = 192 km/h + 176.77 km/h = 368.77 km/h
Horizontal component of the velocity
vxi = 0
vx2 = - 250 km/h x cos(45) = -176.77 km/h
Change in direction of the velocity
θ = arc tan (Vy/Vx)
θ = arc tan(368.77 / -176.77)
θ = -64.4 ⁰
θ = 64.4⁰ north of west.
Acceleration of the trip
a = v/t
v = 408.94 km/h = 113.6 m/s
h = 3 h = 10,800 seconds
a = (113.6 m/s) / ( 10,800 s)
a = 0.0105 m/s²
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