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JEE Advance - Physics (2011 - Paper 1 Offline - No. 17)

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

The phase space diagram for a ball thrown vertically up from ground is
IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 15 English Option 1
IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 15 English Option 2
IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 15 English Option 3
IIT-JEE 2011 Paper 1 Offline Physics - Simple Harmonic Motion Question 15 English Option 4

Explanation

Let the ball of mass m be thrown up with an initial velocity u. Its velocity v and displacement x are related by v2 $$-$$ u2 = $$-$$2gx, where g is the acceleration due to gravity. The momentum (p = mv) is given by

p2 = m2u2 $$-$$ 2m2gx,

which gives

$$p = \pm \sqrt {{m^2}{u^2} - 2{m^2}gx} $$.

At x = 0, the momentum is mu when the ball starts going up and it becomes $$-$$mu when the ball comes back. At the maximum height, x = u2/(2g), the momentum becomes zero.

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