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# A radar station detects a missile approaching from the east. At first contact, the range to the missile is 12,000 ft at 40.0° above the horizon. The missile is tracked for another 123° in the east-west plane, the range at final contact being 25,800 ft; see Fig. Find the displacement of the missile during the period of radar contact.

6 years ago
It is assumed that the position vector of the missile when it was first located is given by vector making an angle  with the horizontal axis pointing east, in the direction of the unit vector , measured counterclockwise.
When the missile was later located, its position vector is given by vector , making an angle with the horizontal axis pointing west, in the direction opposite to the unit vector , measured clockwise.
Also, the magnitude of vector is given by a whereas the magnitude of vector is given by b .
Given:

The figure below shows the position vectors of the missile relative to radar.

Position vector for the missile is given as:

Where ax and ay represents the horizontal and vertical components of the vector.
Also, (refer figure above).
Therefore position vector can be written as:


Substitute the given values of a and  in the vector above to have

Therefore the position vector of the missile when it was first located by the radar is

Position vector for the missile is given as:

Where bx and by represents the horizontal and vertical components of the vector.

Also, (refer figure above).

The value of can be calculated as
 = 180°-123°-40°
= 17°
Therefore position vector can be written as:

The negative sign in horizontal component above highlights the fact that the vector component points in the direction opposite to unit vector .

Substitute the given values of  in the vector above to have

Therefore the position vector of the missile when it was located later by the radar is
.
The displacement vector (say ) of the missile is given as:

If it is assumed that the displacement vector of the missile is given as:

Then on comparing the above vector with vector in equation (1), we have
rx = -15, 480.1 ft
ry = -170.3 ft
The magnitude of displacement of missile is given as:

Substitute the value of rx and ry from above

Rounding off to two significant figures, we have

Therefore the magnitude of displacement vector of the missile is 15,000 .