Two submarines A and B have their routes planned so that their positions at time t hours,0 ≤ t < 20 , would be defined by the position vectors rA = ( 2 4 − 1 ) + t ( − 1 1 − 0.15 ) andrB = ( 0 3.2 − 2 ) + t ( − 0.5 1.2 0.1 ) relative to a fixed point on the surface of the ocean (all lengths arein kilometres).To avoid the collision submarine B adjusts its velocity so that its position vector is now given byrB = ( 0 3.2 − 2 ) + t ( − 0.45 1.08 0.09 ) .

Question

HLPaper 2

Two submarines A and B have their routes planned so that their positions at time t hours,0 ≤ t < 20 , would be defined by the position vectors r_A $= \left( \begin{gathered} \,2 \hfill \\ \,4 \hfill \\ - 1 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 1 \hfill \\ \,1 \hfill \\ - 0.15 \hfill \\ \end{gathered} \right)$ andr_B $= \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.5 \hfill \\ \,1.2 \hfill \\ \,0.1 \hfill \\ \end{gathered} \right)$ relative to a fixed point on the surface of the ocean (all lengths arein kilometres).

To avoid the collision submarine B adjusts its velocity so that its position vector is now given by

r_B $= \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.45 \hfill \\ \,1.08 \hfill \\ \,0.09 \hfill \\ \end{gathered} \right)$ .

Show that the two submarines would collide at a point P and write down the coordinates of P.

[4]

Verified

Solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

r_A=r_{B (M1)}

2 − t= − 0.5t ⇒ t= 4 A1

checking t= 4 satisfies 4+t= 3.2+ 1.2t and − 1 − 0.15t= − 2+ 0.1t R1

P(−2, 8, −1.6) A1

Note: Do not award final A1 if answer given as column vector.

[4 marks]

Find the value of t when submarine B passes through P.

[2]

Verified

Solution

$\left( \begin{gathered} \, - 0.45t \hfill \\ 3.2 + 1.08t \hfill \\ - 2 + 0.09t \hfill \\ \end{gathered} \right) = \left( \begin{gathered} - 2 \hfill \\ \,8 \hfill \\ - 1.6 \hfill \\ \end{gathered} \right)$ M1

Note: The M1 can be awarded for any one of the resultant equations.

$\Rightarrow t = \frac{{40}}{9} = 4.44 \ldots$ A1

[2 marks]

Find the value of t when the two submarines are closest together.

[2]

Verified

Solution

minimum when $\frac{{{\text{d}}D}}{{{\text{d}}t}} = 0$ (M1)

t= 3.83 A1

[2 marks]

Find the distance between the two submarines at this time.

[1]

Verified

Solution

0.511 (km) A1

[1 mark]

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