A rocket is rising vertically at a speed of $300{\text{ m}}{{\text{s}}^{ - 1}}$ when it is 800 m directly above the launch site. Calculate the rate of change of the distance between the rocket and an observer, who is 600 m from the launch site and on the same horizontal level as the launch site.

let x = distance from observer to rocket

let h = the height of the rocket above the ground 

METHOD 1

$\frac{{{\text{d}}h}}{{{\text{d}}t}} = 300{\text{ when }}h = 800$     A1

$x = \sqrt {{h^2} + 360\,000}  = {({h^2} + 360\,000)^{\frac{1}{2}}}$     M1

$\frac{{{\text{d}}x}}{{{\text{d}}h}} = \frac{h}{{\sqrt {{h^2} + 360\,000} }}$     A1

when h = 800

$\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{{{\text{d}}x}}{{{\text{d}}h}} \times \frac{{{\text{d}}h}}{{{\text{d}}t}}$     M1

$= \frac{{300h}}{{\sqrt {{h^2} + 360\,000} }}$     A1

$= 240{\text{ (m}}{{\text{s}}^{ - 1}})$     A1

[6 marks]

METHOD 2

${h^2} + {600^2} = {x^2}$     M1

$2h = 2x\frac{{{\text{d}}x}}{{{\text{d}}h}}$     A1

$\frac{{{\text{d}}x}}{{{\text{d}}h}} = \frac{h}{x}$

$= \frac{{800}}{{1000}}\left( { = \frac{4}{5}} \right)$     A1

$\frac{{{\text{d}}h}}{{{\text{d}}t}} = 300$     A1

$\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{{{\text{d}}x}}{{{\text{d}}h}} \times \frac{{{\text{d}}h}}{{{\text{d}}t}}$     M1

$= \frac{4}{5} \times 300$

$= 240{\text{ (m}}{{\text{s}}^{ - 1}})$     A1

[6 marks]

METHOD 3

${x^2} = {600^2} + {h^2}$     M1

$2x\frac{{{\text{d}}x}}{{{\text{d}}t}} = 2h\frac{{{\text{d}}h}}{{{\text{d}}t}}$     A1A1

when h = 800, x =1000

$\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{{800}}{{1000}} \times \frac{{{\text{d}}h}}{{{\text{d}}t}}$     M1A1

$= 240{\text{ (m}}{{\text{s}}^{ - 1}})$     A1

[6 marks]

METHOD 4

Distance between the observer and the rocket $= {({600^2} + {800^2})^{\frac{1}{2}}} = 1000$     M1A1

Component of the velocity in the line of sight $= \sin \theta \times 300$

(where $\theta =$ angle of elevation)     M1A1

$\sin \theta = \frac{{800}}{{1000}}$     A1

component $= 240{\text{ (m}}{{\text{s}}^{ - 1}})$     A1

[6 marks]

Questions of this type are often open to various approaches, but most full solutions require the application of ‘related rates of change’. Although most candidates realised this, their success rate was low. This was particularly apparent in approaches involving trigonometric functions. Some candidates assumed constant speed – this gained some small credit. Candidates should be encouraged to state what their symbols stand for.