Question

give me a quick answer


[In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O.]
At time t = 0, a bird A leaves its nest, that is located at the point with position vector
(20i - 17j)m, and flies with constant velocity (-6i + 7j)m s~
. At the same time a second
bird B leaves its nest which is located at the point with position vector (-8i + 9j)m and flies with constant velocity (pi + 2pj) ms
, where p is a constant. At time t = 4s, bird B is
south west of bird A.
(a) Find the direction of motion of A, giving your answer as a bearing to the nearest degree.
(3)
(b) Find the speed of B.
(10)

Answer 1

To solve the problem, we can break it down into two parts:

### (a) Finding the direction of motion of bird A

The velocity vector of bird A is given by:

\[
\mathbf{v_A} = -6\mathbf{i} + 7\mathbf{j}
\]

To find the direction of motion, we calculate the angle θ with respect to the positive x-axis (east) using the tangent function:

\[
\theta = \tan^{-1} \left( \frac{\text{opposite}}{\text{adjacent}} \right) = \tan^{-1} \left( \frac{7}{-6} \right)
\]

Calculating this:

\[
\theta = \tan^{-1} \left( -\frac{7}{6} \right) \approx -48.37^\circ
\]

Since this angle is measured clockwise from the positive x-axis (east), we convert it to a bearing:

\[
\text{Bearing} = 360^\circ + (-48.37^\circ) = 311.63^\circ \approx 312^\circ
\]

### Answer to (a):
The direction of motion of A is **312°**.

### (b) Finding the speed of bird B

The position vector of bird A at time \( t \):

\[
\text{Position of A at } t = 4: \quad \mathbf{P_A}(4) = (20 - 6(4))\mathbf{i} + (-17 + 7(4))\mathbf{j} = (20 - 24)\mathbf{i} + (-17 + 28)\mathbf{j} = -4\mathbf{i} + 11\mathbf{j}
\]

The position vector of bird B at time \( t \):

\[
\text{Position of B at } t = 4: \quad \mathbf{P_B}(4) = (-8 + 4p)\mathbf{i} + (9 + 8p)\mathbf{j}
\]

At \( t = 4 \), it is given that bird B is southwest of bird A, which means that:

1. The x-coordinate of B must be less than that of A: \(-8 + 4p < -4\)
2. The y-coordinate of B must be less than that of A: \(9 + 8p < 11\)

From the x-coordinate inequality:

\[
-8 + 4p < -4 \implies 4p < 4 \implies p < 1
\]

From the y-coordinate inequality:

\[
9 + 8p < 11 \implies 8p < 2 \implies p < \frac{1}{4}
\]

Thus, combining these inequalities:

\[
p < \frac{1}{4}
\]

Next, we calculate the speed of bird B. The velocity vector of bird B is:

\[
\mathbf{v_B} = p\mathbf{i} + 2p\mathbf{j}
\]

The speed is the magnitude of the velocity vector:

\[
|\mathbf{v_B}| = \sqrt{p^2 + (2p)^2} = \sqrt{p^2 + 4p^2} = \sqrt{5p^2} = p\sqrt{5}
\]

Providing the final answer for speed, \( p \) can take values below \( \frac{1}{4} \), thus the speed depends on the exact value of \( p \) but must be calculated in the context:

Assuming \( p = \frac{1}{4} \) maximum for maximum speed:

\[
|\mathbf{v_B}| = \frac{1}{4}\sqrt{5} \approx 0.559 \text{ m/s}
\]

### Answer to (b):
The speed of B is **\( p\sqrt{5} \) m/s**, where \( p < \frac{1}{4} \).