Try this problem, useful for Physics Olympiad based on Vertical Motion. The Problem: Vertical Motion A boy is standing on top of a tower of height 85m and throws a ball in the vertically upward direction at a certain speed. If 5.25 secs later he hears the ball hitting the ground, then the speed with […]
Try this problem useful for Physics Olympiad based on Relative motion. The Problem: Relative motion Problem A 40kg boy is standing on a plank of mass 160Kg. The plank originally at rest, is free to slide on a smooth frozen lake. The boy walks along the plank at a constant speed of 1.5m/s relative to […]
Let's solve a beautiful and tricky integral problem. The Problem: Let $$ I$$=$$\int e^x/(e^{4x}+e^{2x}+1) dx$$ $$ J$$=$$ \int e^{-x}/(e^{-4x}+e^{-2x}+1)dx$$. Find the value of (J-I). Solution: $$ I$$=$$\int e^x/(e^{4x}+e^{2x}+1) dx$$ $$J$$= $$\int e^{-x}/(e^{-4x}+e^{-2x}+1)dx$$ Let (e^x)=(z) $$ J-I$$=$$\int\frac{e^x(e^{2x-1})}{e^{4x}+e^{2x}+1}dx$$=$$\int\frac{z^2-1}{z^4+z^2+1}dz $$ $$ =\frac{1}{2}ln\frac{(e^x+e^-x-1)}{(e^x+e^-x+1)}+c$$ (where c is a constant of integration)
A canoe has a velocity of 0.40m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50m/s east relative to the earth. Find the velocity (magnitude and direction ) of the canoe relative to the river. Discussion: We apply the relative velocity relation. The relative velocities are (\vec{v_{CE}}), the canoe […]
Try this problem, useful for Physics Olympiad based on Maximum Height of Object over a Pulley. The Problem: Maximum Height of Object over a Pulley Two objects with masses 5Kg and 2Kg hang 0.6m above the floor from the ends of a cord 6m long passing over a frictionless pulley. Both objects start from rest. […]
Try this beautiful Problem, useful for Physics Olympiad based on Block on a Cart. The Problem: Block on a Cart A block is placed against the vertical front of a cart. What acceleration must the cart have so that block A does not fall? The coefficient of static friction between the block and the cart […]
Try this beautiful problem, useful for Physics Olympiad based on Two Blocks on an Inclined Plane. The Problem: Two blocks of masses (4kg) and (8kg) are connected by a string and slide down a (30^\circ) inclined plane. The coefficient of kinetic friction between the (4Kg) block and the plane is (0.25); that between the (8Kg) […]
Try this problem useful for the Physics Olympiad based on the Position of a Particle. The problem: A particle of mass m is subject to a force $$ F(t)=me^{-bt}$$. The initial position and speed are zero. Find (x(t)). Solution: In the given problem $$ \ddot{x}=e^{-bt}$$ Integrating this with respect to time gives $$ v(t)=-\frac{e^{-bt}}{b}+A $$ […]
Try this problem from TIFR 2013 problem 5 based on Non-Cyclic Subgroup of \(\mathbb{R}\). Question: TIFR 2013 problem 5 True/False? All non-trivial proper subgroups of \((\mathbb{R},+)\) are cyclic. Hint: What subgroups comes to our mind immediately? Discussion: \((\mathbb{Q},+)\) is a subgroup of \((\mathbb{R},+)\). Is \((\mathbb{Q},+)\) a cyclic group? Suppose \((\mathbb{Q},+)\) is cyclic. Then there exists a generator say […]
Let's discuss a beautiful problem useful for Physics Olympiad based on the leaning ladder. The Problem: The Leaning Ladder A ladder leans against a frictionless wall. If the coefficient of friction with the ground is \(\mu\), what is the smallest angle that the ladder can make with the ground and not slip? Solution: Let the […]