Question

derive an expression for finding the velocity of an approaching aeroplane using radar waves

please give a explanation ​

Answer 1

$\huge{\bf{\underline{\overline{\mid{Answer:-}\mid}}}}$

$\huge{\underline{\tt{Derivation:-}}}$

• Let the aeroplane speed speed be v and the speed of light be c.

★ The droppler effect tells you the frequency shift, df of a wave reflecting from a moving object is

df= f0(1-2v/c cos(q))

where q = angle between the object's velocity vector and the propagation vector direction of the radar wave.

Now,

$\implies{ \tt{df = f _{0}(1 - \frac{2v}{c} \cos(q)) }} \\ \implies \: { \tt{ \frac{df}{f _{0} } = 1 - \frac{2v}{c} cosq}} \\ \\ \implies \: { \boxed{ \bf{v = \frac{c( \frac{1 - df}{f _{0}}) }{2 \cos(q) } }}}$

Answer 2

Answer:

vΔn/2n

Explanation:

The radio waves sent from the earth strikes approaching aeroplane. Here the radar is a source which is stationary and the aeroplane is observer which is moving towards the stationery source. We have to determine its velocity.

∴ apparent frequency n' = (v + vₐ/v) n

v = velocity of radar waves and vₐ = velocity of aeroplane.

Now, aeroplane receives waves of frequency n and acts as a source moving towards stationary observer. since on reflection, the frequency does not change, the aeroplane will reflect waves of frequency n₁.

Apparent frequency received by radar is given by,

n₁ = (v - vₐ/v) * n

==> (v - vₐ/v) * (v + vₐ/v)/(v + vₐ/v) * n

==> (1 + vₐ/v)² * n

==> (n₁/n) = (1 + 2vₐ/v) * n

==> (vₐ - v/v) * v

==> vₐ = vΔn/2n

Result:

Velocity of an approaching aeroplane vₐ = vΔn/2n

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