There are two particles of masses m and 2m placed at a distance‘d’ apart on a smooth horizontal surface. Where will the collision occur (with respect to original positions), if they are allowed to move towards each other because of their mutual attraction?a)2d/3b)d/2c)2dd)dCorrect answer is option 'A'. Can you explain this answer?

Question

There are two particles of masses m and 2m placed at a distance&#8216;d&#8217; apart on a smooth horizontal surface. Where will the collision occur (with respect to original positions), if they are allowed to move towards each other because of their mutual attraction?a)2d/3b)d/2c)2dd)dCorrect answer is option 'A'. Can you explain this answer?

Answer

Their collision will occur at their center of mass.
If we consider body of mass 'm' at origin & '2m' at distance "d" from 1st body on X-axis then, center of mass,
X = m₁(x₁)+m₂(x₂)/m₁+m₂
X = m(0)+2m(d)/m+2m
X = 2md/3m
X = 2d/3
Hence A is the correct answer.

Answer

Yes 'A' is correct

Their collision will occur at their center of mass

If we consider body of mass 'm' at origin & '2m' at distance 'd' from 1st body on X-axis then,

center of mass,

x=m1(x1)+m2(x2)/m1+m2

=m(0)+2m(d)/m+2m

=2md/3m

=2d/3

Answer

Collision Point Calculation
To determine where the collision will occur, we need to consider the forces acting on the two particles and analyze their motion.

1. Forces Acting on the Particles:
- There is a mutual attractive force between the two particles due to their masses. This force can be calculated using Newton's law of gravitation: F = G * (m1 * m2) / r^2, where G is the gravitational constant, m1 and m2 are the masses of the particles, and r is the distance between them.
- Since the particles are on a smooth horizontal surface, there is no friction or any other horizontal forces acting on them. Therefore, the only force acting on each particle is the attractive force between them.

2. Initial Motion:
- Initially, the particles are placed at a distance 'd' apart.
- Due to their mutual attraction, they will start moving towards each other.
- The force between them will accelerate each particle in the direction of the other particle.

3. Analysis of Motion:
- As the particles start moving towards each other, their velocities will increase.
- However, the acceleration of each particle will decrease as they get closer to each other.
- At a certain point, the attractive force between them will be equal to the inertia of each particle, causing them to decelerate and eventually come to a stop.
- After this point, the attractive force will dominate, causing the particles to accelerate in the opposite direction.
- The particles will continue to move away from each other until they come to a stop again due to the attractive force being equal to their inertia.
- This process will continue until the particles collide.

4. Determining the Collision Point:
- Considering the symmetry of the problem, the collision point will be equidistant from the original positions of the particles.
- Let's assume the collision point is at a distance 'x' from each particle's initial position.
- At this point, the attractive force between the particles will be equal to the inertia of each particle.
- Applying Newton's law of gravitation, we can write the equation: G * (m * 2m) / (x + x)^2 = m * a, where 'a' is the acceleration.
- Simplifying the equation, we get: 4Gm^2 / 4x^2 = ma.
- Canceling out m and rearranging, we obtain: x = d/2.
- Therefore, the collision will occur at a distance of d/2 from each particle's initial position.

Answer Explanation:
The correct answer is option 'B' (d/2). This means that the collision will occur at a distance of d/2 from each particle's original position.

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