A telescope gives a clear view of a distant object when the angular displacement between the edges of the object is at least 9.7 × 10-6 rad.

Question 2

(a) Define the radian. [2]

(b) A telescope gives a clear view of a distant object when the angular displacement between the edges of the object is at least 9.7 × 10^-6 rad.

(i) The Moon is approximately 3.8 × 10⁵ km from Earth.

Estimate the minimum diameter of a circular crater on the Moon’s surface that can be

seen using the telescope. [2]

(ii) Suggest why craters of the same diameter as that calculated in (i) but on the surface of Mars are not visible using this telescope. [2]

Reference: Past Exam Paper – June 2014 Paper 42 Q7

Solution:

(a) The radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius.

(b)

(i)

{Since the angular displacement is very small (9.7 × 10⁻⁶ rad), the distance from the edge of the moon to the telescope (r) can be approximated to be equal to the (horizontal) distance of the moon from the telescope on Moon (3.8 × 10⁵ km).

180° correspond to π (which is about 3.14) rad. You can convert the angular displacement given into degrees to have a better idea how small it is.

Length of arc = rθ      

Similarly, the length of the arc formed can be approximated to be equal to the diameter of the moon since the angular displacement is too small.

That’s why the question says ‘estimate’.}

Length of arc = distance × angle       

Minimum diameter of circular crater = 3.8×10⁵ × 9.7×10^-6 = 3.7 km

(ii) Mars is (much) further from Earth / away. So, the angle (at the telescope is much) smaller.