Determine the magnitude and direction of the combined momentum of the moving spheres shown in the figure by drawing a vector diagram. $m_1=2 \mathrm{kg}$, $m_3=3 \mathrm{kg}$, $v_1=6 \mathrm{m}/\mathrm{s}$, $v_2=4\mathrm{m}/\mathrm{s}$. 

 

 

The velocities of two bodies(masses $5 \mathrm{kg}$ and $10 \mathrm{kg}$) in the $+x$ and the $+y$ directions are $4 \mathrm{m}/\mathrm{s}$ and $2 \mathrm{m}/\mathrm{s}$ respectively. They collide and stick together. What is the final velocity?

A metallic block of mass $2 \mathrm{kg}$ is resting on a frictionless plane. It is struck by a jet releasing water at a rate of $1 \mathrm{kg}/\mathrm{s}$ and at a speed of $5 {\mathrm{ms}}^{-1}$. Find the initial acceleration of the block.

The mass of a rocket is $1.3\times {10}^4 \mathrm{kg}$. Its engine exerts on it an initial thrust of $2.6\times {10}^5 \mathrm{N}$ in the upward direction. Calculate the initial acceleration attained by the rocket$\left(g=9.8 \mathrm{m}/{\mathrm{s}}^2\right)$.

In a rocket fuel is burnt at the rate of $100 \mathrm{kg}/\mathrm{s}$ and burnt gases rush at a speed of $5\times {10}^4 \mathrm{m}/\mathrm{s}$. Find the thrust of the rocket ignoring gravity.

In a rocket, the mass of the fuel is $90\%$ of the total mass. The exhaust gases are ejected at a speed of $2 {\mathrm{kms}}^{-1}$. Find the maximum speed attained by the rocket. Neglect gravity and air resistance and take the initial speed of the rocket to be zero. Given : ${\log }_{e}10=2.3$

Based upon the following paragraph answer the following multiple choice question:

Paragraph:

Phase space diagrams are useful tools for 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 the horizontal axis and momentum is plotted along the vertical axis. The phase space diagram is$x\left(t\right)$ vs. $p\left(t\right)$ 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 in a straight line as shown in the figure. We use the sign convention in which opposition and momentum upwards(or to right) are positive and downwards (or to left) are negative.

The phase space diagram for a ball thrown in  the vertically upward direction is:

Phase space diagrams are useful tools in analysing 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$x\left(t\right)$ vs. $p\left(t\right)$ 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 in a straight line as shown in the figure. We use the sign convention in which position and momentum upwards(or to right) are positive and downwards (or to left) is negative.

The phase space diagram for the simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $E_1$ and $E_2$ are the total mechanical energies respectively. Then

Based upon the following paragraph answer the following multiple choice question:

Paragraph:

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$x\left(t\right)$ vs. $p\left(t\right)$ 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 in a straight line as shown in the figure. We use the sign convention in which position and momentum upwards(or to right) are positive and downwards (or to left) is negative.

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is: