- In a certain region a hill is described by the shape z(x,y) = (1/50)x
^{4}+ y^{2}– xy – 3y, where the axes x and y are in the horizontal plane and axis z points vertically upward. If î, ĵ and k̂ are unit vectors along x, y and z, respectively, then at point x = 5, y = 10 the unit vector in the direction of the steepest slope of the hill be:- î
- ĵ
- k̂
- î + ĵ + k̂

Answer:- ĵ

z(x,y) = (1/50)x^{4}+ y^{2}– xy – 3y

⇒ ∂z/∂x = (2/25)x^{3}– y & ∂z/∂y = 2y – x -3

At point (5,10), ∂z/∂x = 0 & ∂z/∂y = 12

Therefore, ∇z = (∂z/∂x)î + (∂z/∂y)ĵ = 12ĵ

⇒ unit vector in the direction of the steepest slope of the hill is ĵ

The gradient of p, ∇p, at a given point in space is defined as a vector such that:- Its magnitude is the maximum rate of change of p per unit length of the coordinate space at the given point
- Its direction is that of the maximum rate of change of p at the given point.

- An aircraft is cruising at an altitude of 9 km. The free-stream static pressure and density at this altitude are 3.08 × 10
^{4}N/m^{2}and 0.467 kg/m^{3}respectively. A pitot tube mounted on the wing senses a pressure of 3.31 × 10^{4}N/m^{2}. Ignoring compressiblity effects, the cruising speed of the aircraft is approximately- 50 m/s
- 100 m/s
- 150 m/s
- 200 m/s

Answer:- 100 m/s

Pitot tube senses total pressure.

p_{0}= p_{∞}+ 0.5ρ_{∞}v_{∞}^{2}

⇒3.31 × 10^{4}= 3.08 × 10^{4}+ (0.5)(0.467)v_{∞}^{2}

⇒v_{∞}= 99.25 m/s ~ 100 m/s

# GATE 2010 AEROSPACE ENGINEERING (Q 31-40)

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