Two identical particles, each of mass , are coasting in free space along the same path, one in front of the other by . At the instant their separation distance has this value, each particle has precisely the same velocity of role="math" localid="1663837344267" . What are their precise velocities when they are 2.0 m apart?
The trailing particle is moving at and the leading particle moves at the velocity .
The mass of each particle is .
The initial distance between the particle is
The final distance between the particles is
The formula for the velocity of the centre of mass is given by
Here, , therefore
As the two particles are approaching at the same speed (say ) with respect to the centre of mass towards each other. This implies that the velocity of the trailing particle increases by and the velocity of the leading particle decreases by. Therefore,
Thus, it is proved that the velocity of the centre of mass remains unchanged.
Applying conservation of energy of the system in the CoM frame,
Inserting the values for and mass of the particle; initial and final distance from the given data, we get
Hence, in the CoM frame, both particles are moving at speed toward each other. And in the lab frame, the trailing particle is moving at and the leading particle moves at velocity .
Two spheres having masses M and 2M and radii R and 3R, respectively, are simultaneously released from rest when the distance between their centers is 12R. Assume the two spheres interact only with each other and we wish to find the speeds with which they collide. (a) What two isolated system models are appropriate for this system? (b) Write an equation from one of the models and solve it for , the velocity of the sphere of mass M at any time after release in terms of , the velocity of 2M. (c) Write an equation from the other model and solve it for speed in terms of speed when the spheres collide. (d) Combine the two equations to find the two speeds and when the spheres collide.
A system consists of three particles, each of mass , located at the corners of an equilateral triangle with sides of . (a) Calculate the potential energy of the system. (b) Assume the particles are released simultaneously. Describe the subsequent motion of each. Will any collisions take place? Explain.
A spacecraft is approaching Mars after a long trip from the Earth. Its velocity is such that it is traveling along a parabolic trajectory under the influence of the gravitational force from Mars. The distance of closest approach will be above the Martian surface. At this point of closest approach, the engines will be fired to slow down the spacecraft and place it in a circular orbit above the surface. (a) By what percentage must the speed of the spacecraft be reduced to achieve the desired orbit? (b) How would the answer to part (a) change if the distance of closest approach and the desired circular orbit altitude were instead of ? (Note: The energy of the spacecraft–Mars system for a parabolic orbit is .)
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