Answer

Verified

55.2k+ views

**Hint**The given particles are in elastic collision. The initial and final velocities, mass of the particle, the area of the metal plate, and the volume density of the gas is provided. To solve this question we must know the concepts of elastic collision and the collision of particles in one and two dimensions. We know that force can be determined by change in momentum. Use these concepts to find the answer.

**Complete step by step solution**

The mass of gas particles is ${\text{m = 1}}{{\text{0}}^{{\text{ - 26}}}}{\text{kg}} $

Velocity before hitting the metal pate (before collision), ${{\text{v}}_{\text{0}}}{\text{ = 5m/s}} $

Velocity after collision, ${\text{v = 2m/s}} $

Volume density of the gas particles is ${\text{n = 1}}{{\text{0}}^{{\text{25}}}}{\text{ per }}{{\text{m}}^3} $

Area of the beam, ${\text{A = 1}}{{\text{m}}^{\text{2}}} $

The external force, F=? (The force required to move the plate with a constant velocity, ${\text{v = 2m/s}} $)

Here the force required to move the plate is determined by the change in the momentum of the particle on the given area of the metal.

$ \Rightarrow {\text{F = }}\Delta {\text{PA }} \to {\text{1}} $

F is the force.

$\Delta P $ is the change in the momentum

We know that Momentum is the product of mass and velocity.

Then, the equation 1 becomes,

$ \Rightarrow {\text{F = m}}\dfrac{{d{v_0}}}{{dt}}{\text{nA}} - {\text{m}}\dfrac{{dv}}{{dt}}{\text{nA}} $ (mass is constant)

$ \Rightarrow {\text{F = mnA}}\left( {\dfrac{{d{v_0}}}{{dt}} - \dfrac{{dv}}{{dt}}} \right) $

Differentiate

\[ \Rightarrow {\text{F = mnA}}\left( {{v_0}^2 - {v^2}} \right)\]

\[ \Rightarrow {\text{F = mnA}}{\left( {{v_0} + v} \right)^2} - {\left( {{v_0} - v} \right)^2}\]

m is the mass

${v_0} $ is the initial velocity

v is the final velocity

n is the volume density of the gas particles

The external force, F=?

\[ \Rightarrow {\text{F = 2}} \times {\text{1}}{{\text{0}}^{ - 26}} \times {10^{25}} \times 1 \times \left( {{{\left( {5 + 2} \right)}^2} - {{\left( {5 - 2} \right)}^2}} \right)\]

\[ \Rightarrow {\text{F = 2}} \times {\text{1}}{{\text{0}}^{ - 26}} \times {10^{25}} \times 1 \times \left( {49 - 9} \right)\]

\[ \Rightarrow {\text{F = 2}} \times {\text{1}}{{\text{0}}^{ - 26}} \times {10^{25}} \times 1 \times 40\]

\[ \Rightarrow {\text{F = 8N}}\]

**The external force, F required to move the plate is \[{\text{F = 8N}}\]**

**Hint**Collision occurs when two objects hit one another. When the object collides with each other its momentum is conserved for an isolated system. The principle of conservation of momentum says that the momentum before collision is equal to the momentum after the collision. The difference between a one dimensional and two dimensional collision is in one dimensional collision, change in velocities of the particles occurs only in one direction so, conservation of momentum is in one direction only whereas in two dimensional, change in velocities of the particles occurs in two direction so, conservation of momentum is in two direction.

Recently Updated Pages

Why do we see inverted images in a spoon class 10 physics JEE_Main

A farsighted man who has lost his spectacles reads class 10 physics JEE_Main

State whether true or false The outermost layer of class 10 physics JEE_Main

Which is not the correct advantage of parallel combination class 10 physics JEE_Main

State two factors upon which the heat absorbed by a class 10 physics JEE_Main

What will be the halflife of a first order reaction class 12 chemistry JEE_Main

Other Pages

Electric field due to uniformly charged sphere class 12 physics JEE_Main

The resultant of vec A and vec B is perpendicular to class 11 physics JEE_Main

A train starts from rest from a station with acceleration class 11 physics JEE_Main

The nitride ion in lithium nitride is composed of A class 11 chemistry JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main