Question

You are analysing data collected by a rover on the moon. You may assume that the rover starts at the origin (0,0,0), and that each coordinate it transmits contains an East-West component, a North-South component and an altitude component, relative to the origin. Let positive movement in the i direction be East, positive movement in the j direction be North, and positive movement in the k direction be increasing altitude. All values are in meters. Assume the moon is flat (neglect its curvature). The rover relays the following four coordinates in order, as it descends into a crater. Coordinates are issued each time the rover changes direction. As such, you may assume that each trajectory between these points is a straight line:
O : (0, 0, 0).
P₁: (2000, 5000, -500).
P₂: (3000, 8000, -600).
P₃: (6000, 9000, z).

Unfortunately, the transmission of the fourth coordinate is corrupted, and does not have an altitude component. For now, you label this component as z.

(a) Determine the displacement vectors describing the rover's straight-line trajectory between each of their transmissions. You may keep z as an unknown.

(b) Upon reaching P₂, the rover's battery has drained and requires recharging. Assuming the battery was initially at full charge, calculate the approximate range of the rover from one battery charge (i.e. what distance can it travel before requiring a recharge).

(c) For communication with the rover, mission control needs to point their antenna (located at O) towards the rover. Ignoring the altitude component (i.e. treat the problem as two-dimensional), calculate the angle that the antenna must be rotated to point towards P₂ if it is currently pointing at P₁.

(d) Based on the time taken to send and receive communications, you calculate that at P₃ the rover must be 10,850 meters from O. Determine z, the altitude component of the co-ordinate P₃. You may assume that P₃ is the lowest point in the crater.

Answer 1

a) The displacement vector from point A(x1,y1,z1) to B(x2,y2,z2) in 3 dimensions is given by

R_AB = (x2-x1) + (y2-y1) + (z2-z1).

Thus,

R_OP1 = (2000-0) + (5000-0) + (-500-0). = 2000 + 5000 - 500.

R_P1P2 = (3000-2000) + (8000-5000) + (-600--500). = 1000 + 3000 - 100.

R_P2p3 = (6000-3000) + (9000-8000) + (z-600-0). = 3000 + 1000 + (z-600).

b) Distance between two points is given by

D(R)=

The rover travelled from O to P1 and from P1 to P2 with its battery.

So, D(R_OP1) =

D(R_P1P) =

So, total distance before recharge = 5408.33+3178.05 = 8586.38 m.

c) In two dimension, the angle between a point and origin is given by tan() = Y/X

For the point P1, Y = 5000, X = 2000

tan() = 5000/2000 = 2.5.

= 68.2 degrees

For the point P2, Y = 8000, X = 3000

tan() = 8000/3000 = 2.67

= 69.47 degrees

So, the change in angle to be made = 69.47-68.2 = 1.27 degrees to the anti- clockwise direction.

d) We have, So, D(R_OP3) = 10850 m.

Substituting this in the equation for D,

(since the item is in the crater, negative root should be taken.)