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Motion in one dimension Problem

The relation between time t and distance x is t = αx+ βx, where α & β are constants, then the retardation is

Sriyathmika asked

Asnwer

Given  t = αx2 + βx

Differentiate both sides with respect to time (t)
dt/dt = 2αx.dx/dt + β.dx/dt
1 = 2αxv + βv
v.(2αx + β) = 1
(2αx + β) = 1/v
Again differentiate both side with respect to time (t)
2α.dx/dt = -v -2 . dv/dt
2αv = v -2 . acceleration
acceleration = -2αv 3
Hence, retardation= 2αv 3

CRUISE LIFT VEHICLE

A vehicle cruises at a speed of 3 km/s at an altitude of 25 km. It has a mass of 100,000 kg, and it has a lifting surface area of 300  . Take the radius of the Earth to be 6400 km, and the value of ‘g’ to be 9.8 ms-1 . The lift to drag ratio is 5.
Both answers should be given in N.

How much lift must it generate?
How much thrust must the propulsion system generate?

(JH)

Hypersonic Equivalence Principle

A piston moves along a tube containing air at an initial sound speed of 330 m/s. When the piston velocity is 250 m/s, it drives a shock wave which propagates at a velocity of 500 m/s. When the piston velocity is 100 m/s, it drives a shock at 392 m/s.

Use the hypersonic equivalence principle to calculate the shock angles (in degrees) on a flat plate:

At an incidence of 6 degrees and a Mach number of 7.2
At an incidence of 2 degrees and a Mach number of 21.7

Then calculate:

The incidence degrees required to produce a shock angle of 9.5 degrees at a Mach number of 7.2

The incidence () required to produce a shock angle () of 3.2 degrees at a Mach number of 21.7

 

(JH)

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