The site from which an airplane takes off is the origin. The x axis points east; the y axis points straight up. The position and velocity vectors of the plane at a later time are given by .

The magnitude, in meters, of the plane's displacement from the origin is
A)9.14 × 103.
B)1.61 × 104.
C)1.84 × 104.
D)9.14 × 103t.
E)1.61 × 104t.

Question

The site from which an airplane takes off is the origin. The x axis points east; the y axis points straight up. The position and velocity vectors of the plane at a later time are given by ​  .
​
The magnitude, in meters, of the plane's displacement from the origin is
A)9.14 × 103. 
B)1.61 × 104. 
C)1.84 × 104. 
D)9.14 × 103t. 
E)1.61 × 104t.

Quizplus · Accepted Answer

To find the displacement vector, we need to integrate the velocity vector over time.

∫(8.00i+6.00j−5.00k)dt = (8.00t)i + (6.00t)j − (2.50t²)k

To find the magnitude of this vector, we take the square root of the sum of the squares of its components:

|displacement| = sqrt((8.00t)² + (6.00t)² + (2.50t²)²)

|displacement| = sqrt(64t² + 36t² + 6.25t⁴)

|displacement| = sqrt(70.25t⁴ + 100t²)

|displacement| = t*sqrt(70.25t² + 100)

At the final time t = 2 min = 120 s, we have:

|displacement| = 120*sqrt(70.25*120² + 100) 
|displacement| ≈ 18,357.8 m

Therefore, the correct answer is (C) 1.84 × 10^4 meters.

The Site from Which an Airplane Takes Off Is the Origin