Physics Guru

Evaluate, $I = \int {\frac{{{x^3} + 2x + 1}}{{{x^2} + 1}}} dx$ We have, $I = \int {\frac{{{x^3} + x + x + 1}}{{{x^2} + 1}}} dx$ = $ \int {\frac{{x({x^2} + 1) + x + 1}}{{{x^2} + 1}}} dx$ $ = \int {x + \frac{{x + 1}}{{{x^2} + 1}}dx} $ $ = \int {xdx + \int {\frac{x}{{{x^2} + 1}}} } dx + \int {\frac{{dx}}{{1 + {x^2}}}} $ $ = \frac{{{x^2}}}{2} + \frac{1}{2}\int {\frac{{2x}}{{{x^2} + 1}}} dx + {\tan ^{ - 1}}x + C$ $ = \frac{{{x^2}}}{2} + \frac{1}{2}\ln ({x^2} + 1) + {\tan ^{ - 1}}x + C$

Evaluate, $I = \int\limits_0^\infty {\frac{{dx}}{{{{(1 + {x^2})}^5}}}} $ $Let,x = \tan \theta $ $\therefore dx = {\sec ^2}\theta d\theta $ $\therefore I = \int\limits_0^{\frac{\pi }{2}} {\frac{{{{\sec }^2}\theta }}{{{{(1 + {{\tan }^2}\theta )}^5}}}} d\theta $ $\therefore I = \int\limits_0^{\frac{\pi }{2}} {\frac{{{{\sec }^2}\theta }}{{{{({{\sec }^2}\theta )}^5}}}} d\theta $ $\therefore I = \int\limits_0^{\frac{\pi }{2}} {{{\cos }^8}\theta d\theta } $ $\therefore I = \frac{{7 \cdot 5 \cdot 3 \cdot 1}}{{8 \cdot 6 \cdot 4 \cdot 2}} \times \frac{\pi }{2} = \frac{{7 \cdot 5}}{{8 \cdot 2 \cdot 4 \cdot 2}} \times \frac{{22}}{{7.2}}$ $\therefore I = \frac{{55}}{{128}}$

Q: A bird while flying takes a right turn, where does it get the centripetal force from? A: The centripetal force is provided by the bird’s wings as it changes the direction of its flight. As the bird turns to the right, it must adjust the angle of its wings to produce a component of the force towards the center of the turn.

Among the following, the correct statement(s) for electrons in an atom is(are)  (A) Uncertainty principle rules out the existence of definite paths for electrons. (B) The energy of an electron in 2s orbital of an atom is lower than the energy of an electron that is infinitely far away from the nucleus. (C) According to Bohr’s model, the most negative energy value for an electron is given by n = 1, which corresponds to the most stable orbit. (D) According to Bohr’s model, the speed of electrons increases with increase in values of n.

 $\frac{{{\rm{9 \: digit \: number \: nnnnnnnnn}}}}{{n + n + n + n + n + n + n + n + n}} = $ (A) not an integer (B) 12345678 (C) depends on n (D) 12345679

How many 3-digit hexadecimal numbers can be formed if each digit can be any of the characters from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}, with the restriction that the starting/left digit cannot be 0?

Plane A flies at a constant speed of 300 km/h. Plane B is 225 km behind plane A and flies at an initial speed of 420 km/h, but due to fuel limitations, its speed decreases linearly by 30 km/h every hour. Assume both planes fly in horizontal path at safe distance from each other.  (A) They will cross each other at t = 3 hrs. only (B) They will cross each other at t = 5 hrs. only (C) They will cross each other two times (D) They will never cross each other

Which phenomenon of light is responsible for rain blocking the vision? (A) Total internal reflection (B) Dispersion (C) Interference (D) Scattering

A commercial plane experiences a complete loss of all engine power at an altitude of 35,000 feet (1 km = 3280.84 feet) and an airspeed of 450 knots (1 knot = 1.852 km/h). The Ram Air Turbine (RAT) is immediately deployed and the pilots gets ready for emergency landing. The RAT is designed to generate 20 kVA of electrical power with power generation efficiency of 85%. Power factor is 0.8 for a typical aircraft electrical system. The aircraft's essential systems require a continuous power supply of 13.5 kW. The aircraft has an average glide ratio of 15:1 (meaning for every 15 units of horizontal distance covered, it loses 1 unit of altitude). For how many minutes can the RAT continuously power the essential systems if the aircraft maintains a constant glide until it reaches the ground level for an emergency landing? (Ignore any potential battery backup before touchdown and also ignore RAT after touchdown for this calculation).

A spherical neutron star has density equal to that of the nucleus. The radius of the neutron star having mass $5.0 \times 10^{30}$ kg is (A) $\sqrt 2$ km (B) $\sqrt 3$ km (C) $\sqrt 2 \times 10^4 $ m (D) $\sqrt 3 \times 10^4 $ m

Consider an aeroplane covering the entire runway 3.5 km long before take-off. The initial speed of the plane is zero and the final speed just before take-off is 174 knots (1 knot = 1.852 kmph). Assuming the acceleration to be constant, how much time is taken to cover the runway?

It is a common practice in India to keep drinking water in earthen pot in summer so as to cool it. What makes it work?

When brakes are applied to stop the car, how is the car stopped as a system cannot apply force on itself.

An astronaut sitting in a chair in a satellite feels weightless. However, Moon is also a satellite and the astronaut does not feel weightless there. Are laws of Physics different for man-made satellite and the God made satellite?