At time t = 0, a small boat is at a buoy travelling due east with speed 2 ms−1

. The boat
experiences a constant acceleration of (– 0.4i + 0.1j) ms−2
. The unit vectors i and j are
directed east and north respectively.
(a) Write down the initial velocity of the boat. (1)
(b) Find an expression for the displacement of the boat at time t seconds. (2)
(c) By considering the i component of the displacement vector of the boat, find the
time when the boat is due north of the buoy. (3)
(d) Find an expression for the velocity of the boat at time t seconds and hence find the
time when the boat is travelling northeast from the buoy.

(a) The initial velocity of the boat can be obtained from the information given: the boat is travelling due east with a speed of 2 m/s. Since there is no motion in the vertical direction, the initial velocity can be written as v0 = 2i m/s.

(b) To find the expression for the displacement of the boat at time t seconds, we need to integrate the velocity function with respect to time. Since the acceleration is constant, we can use the following equation:

∫(v - v0) dt = ∫a dt

Integrating each component separately, we get:

∫dx = ∫(0.4t - 0) dt
x = 0.2t^2 + C1

∫dy = ∫(0.1t - 0) dt
y = 0.05t^2 + C2

Here, C1 and C2 are integration constants.

The displacement vector of the boat at time t seconds can be written as r = x i + y j, where x and y are the expressions we found above:

r = (0.2t^2 + C1) i + (0.05t^2 + C2) j

(c) To find the time when the boat is due north of the buoy, we need to consider the i component of the displacement vector. At this time, the x-distance will be zero. So, we equate the expression for x to zero:

0.2t^2 + C1 = 0

Solving this equation, we get C1 = 0. Therefore, the i component of the displacement equation is:

x = 0.2t^2

To find the time when the boat is due north, y component should be maximum. Since y = 0.05t^2 + C2, maximum y occurs when t = 0. Therefore, the boat is due north of the buoy at time t = 0 seconds.

(d) The velocity of the boat at time t seconds can be obtained by taking the derivative of the displacement equation with respect to time:

v = (dx/dt) i + (dy/dt) j

Differentiating each component separately, we get:

dx/dt = 0.4t
dy/dt = 0.1t

Therefore, the velocity vector of the boat at time t seconds is:

v = (0.4t) i + (0.1t) j

To find the time when the boat is traveling northeast, both i and j components of the velocity should be positive. This occurs when t > 0. Therefore, the boat is traveling northeast from the buoy for any time t greater than zero.