Ultrasound is utilized in measuring blood flow non-invasively using a phenomenon referred to as Doppler Effect. a. Derive the expression of the Doppler shift. [10] b. A horseshoe bat flies towards a moth at vb = 9 m/s, while the moth flies towards the bat at vm = 8 m/s. The bat emits ultrasonic waves of frequency fbe that reflect from the moth back to the bat with frequency fbd. The bat adjusts the emitted fbe so that fbd = 83 kHz, its optimal hearing frequency. i. Find the frequency fm heard and reflected by the moth. [2] ii. Calculate the frequency fbe emitted by the bat. [2] c. In several cases, a nearby star has been found to have a large planet orbiting about it, although the planet could not be seen. Using the ideas of a system rotating about its centre of mass and of the Doppler shift for light (which is in several ways similar to the Doppler Effect for sound), explain how an astronomer could determine the presence of the invisible planet