Engineering Mechanics

(^412) „„„„„ A Textbook of Engineering Mechanics
Now in order to find the least distance between the two bodies, first of all, we have to find out
the relative velocity of one body with respect to the other. So let us draw the relative velocity diagram
XMRN for the two bodies, as usual in which XR represents the relative velocity as shown in
Fig. 19.12 (b).
Let us superimpose the least distance diagram on the relative velocity diagram as shown in
Fig. 19.12 (b) and as discussed below :

1. First of all, extend the relative velocity line XR to C.


2. Now cut off XP equal to the distance between the two bodies at any given instant to the
   scale.


3. From P, draw PQ perpendicular to XC. The length PQ will give the least distance between
   the two bodies to the scale.


4. The time elapsed before the two bodies are least distance apart is given by the relation :

Length to the scale

=

Relative velocity of two bodies

PQ

Note. A little consideration will show that the two bodies will go on coming nearer to each

other till one of the body reaches Q. Beyond this point, the two bodies will go away from each other.

This happens as the least distance between a point and a line is the perpendicular from the point to the

line.

Example 19.8. A ship sailing East with a velocity of 15 km. per hour, passes a certain point

at noon ; and a second ship, sailing North with the same velocity passes the same point at 1.30 P.M.

At what time are they closest together and what is the distance between them?

Solution. Given. Velocity of first ship = 15 km.p.h. (East) and velocity of second ship

= 15 km.p.h. (North).

Fig. 19.13.

Closest distance between the two ships

First of all, let us draw the actual velocity diagram of the two ships as shown in Fig. 19.13 (a).

Now draw the relative velocity diagram as shown in Fig. 19.13 (b) and as discussed below :

1. First of all, draw the East, West, North and South lines meeting at X.


2. Since the first ship is sailing East, therefore draw XL representing the actual direction of
   the first ship.


3. Now cut off XM equal to 15 km to some suitable scale on the opposite direction of the
   actual motion of the first ship (i.e. towards West).