Theory question 116
Vectors(a) Two ships M and N, moving with constant velocities, have position vectors (3i + 7j) and (4i + 5j) respectively. If the velocities of M and N are (5i + 6j) and (2i + 3j) and the [distances] of the ships after t seconds are in metres, find (i) MN (ii) |MN|, when t = 3 seconds.
(b) A particle is acted upon by forces F1 = 5i + pj, F2 = qi + j, F3 = 2pi + 3j and F4 = −4i + qj, where p and q are constants. If the particle remains in equilibrium under the action of these forces, find the value of p and q.
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(a) Position at time t: M = (3 + 5t)i + (7 + 6t)j, N = (4 + 2t)i + (5 + 3t)j
(i) MN = N − M = (1 − 3t)i − (2 + 3t)j; at t = 3, MN = −8i − 11j
(ii) |MN| = √(64 + 121) = √185 ≈ 13.6 m
(b) Sum of forces = 0
i: 5 + q + 2p − 4 = 0, so 2p + q = −1
j: p + 1 + 3 + q = 0, so p + q = −4
p = 3, q = −7
Position at time t = starting position + t × velocity; in equilibrium the i and j parts each add to zero.