Inelastic collisions

Collisions in 1D

Problem:

A particle of mass m1 and velocity u1 collides with a particle of mass m2 at rest.  The two particles stick together.  What fraction of the original kinetic energy is lost in the collision?  (Simplify the answer as much as possible.)

Solution:

Problem:

A particle of mass m traveling with (non-relativistic) velocity u1 makes a head-on collision with a second particle of mass M, which is at rest in the laboratory.  If the collision is completely inelastic, what fraction of the original kinetic energy remains after the collision?

Solution:

Problem:

A 36 g bullet with a speed of 350 m/s strikes a 8 cm thick fence post.  The bullet is retarded by an average force of 3.6*103 N while traveling all the way through the board.
(a)   What speed does the bullet have when it emerges?
(b)   How many such boards could the bullet penetrate?

Solution:

Problem:

Board A is placed on board B as shown.  Both boards slide, without moving with respect to each other, along a frictionless horizontal surface at a speed v.  Board B hits a resting board C "head-on."  After the collision, boards B and C move together, and board A slides on top of board C and stops its motion relative to C in the position shown on the diagram.  What is the length of each board?  All three boards have the same mass, size, and shape.  It is known there is no friction between boards A and B; the coefficient of kinetic friction between boards A and C is µk.

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Solution:

Problem:

An hour glass sits on a scale.  Initially all the sand (mass m) in the glass (mass M) is held in the upper reservoir.  At t = 0, the sand is released.  If it exits the upper reservoir at a constant rate dm/dt = -λ, draw (and label quantitatively) a graph showing the reading of the scale at all times t > 0.

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Solution:

Problem:

A 15.2 g bullet hits a 0.463 kg block from below.  The initial speed of the bullet is 624 m/s and it emerges from the block at 131 m/s.
(a)  How high does the block rise?
(b)  If the block is 2.34 cm thick, estimate the average force on the block.  Assume that the bullet passes completely through before the block moves appreciably.

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Solution:


Collisions in 2D

Problem:

A 90 kg fullback running east with a speed of 5 m/s is tackled by a 95 kg opponent running north with a speed of 3 m/s.  If the collision is perfectly inelastic, calculate the speed and the direction of the players just after the tackle.

Solution:

Problem:

After a completely inelastic collision between two objects of equal mass, each having initial speed v, the two move off together with speed v/3.  What was the angle between their initial directions?

Solution:

Problem:

The mass of the blue puck is 20% greater than the mass of the green one.  Before colliding, the pucks approach each other with equal and opposite momenta, and the green puck has an initial speed of 10 m/s.  Find the speed of the pucks after the collision, if half the kinetic energy is lost during the collision.

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Solution:

Problem:

A wooden block of mass M hangs on a massless rope of length L.  A bullet of mass m collides with the block and remains inside the block.  Find the minimum velocity of the bullet so that the block completes a full circle about the point of suspension.

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Solution:

Problem:

A car of total mass M1 = M and velocity v1 makes a totally inelastic collision at time t = 0 with a second car of mass M2 = 2M at rest.  Before the collision a point object of mass m << M was sitting at the bottom of a frictionless spherical cavity of radius r embodied inside the first car.  For what range of velocities v1 will the small mass lose contact with the surface of the cavity?

Solution:

Problem:

A brick is thrown (from ground level) with speed V at an angle θ with respect to the (horizontal) ground.   Assume that the long face of the brick remains parallel to the ground at all times, and that there is no deformation in the ground or the brick when the brick hits the ground.  If the coefficient of friction between the brick and the ground is μ what should θ be so that the brick travels the maximum total horizontal distance before finally coming to rest?

Solution:

Problem:

A marble bounces down stairs in a regular manner, hitting each step at the same place and bouncing the same height above each step. 

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The stair height equals its depth (tread = rise) and the coefficient of restitution ε is given.  Find the necessary horizontal velocity and bounce height.  
(The coefficient of restitution is defined as ε = -vf/vi, where vf and vi are the vertical velocities just after and before the bounce. respectively).