- In a certain region a hill is described by the shape z(x,y) = (1/50)x4 + y2– 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)x4 + y2 – xy – 3y
⇒ ∂z/∂x = (2/25)x3 – 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 × 104 N/m2 and 0.467 kg/m3 respectively. A pitot tube mounted on the wing senses a pressure of 3.31 × 104 N/m2. 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.
p0 = p∞ + 0.5ρ∞v∞2
⇒3.31 × 104 = 3.08 × 104 + (0.5)(0.467)v∞2
⇒v∞ = 99.25 m/s ~ 100 m/s
Sura's Blog 