## Motion in one dimension Problem

# The relation between time t and distance x is t = αx^{2 }+ βx, where α & β are constants, then the retardation is

Sriyathmika asked

**Asnwer**

Given t = αx^{2} + βx

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}

^{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**)