A moving particle suffers elastic collision in one direction with another particle , originally at rest , having mass(i) x times (ii) 1/x times that of first particle . Prove that the fraction of K.E. transferred from first to second ball is same in both the cases and obtain an expression for the fraction
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Answer:
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Explanation:
When objects collide, they can either stick together or bounce off one another, remaining separate. In this section, we’ll cover these two different types of collisions, first in one dimension and then in two dimensions.
In an elastic collision, the objects separate after impact and don’t lose any of their kinetic energy. Kinetic energy is the energy of motion and is covered in detail elsewhere. The law of conservation of momentum is very useful here, and it can be used whenever the net external force on a system is zero. Figure 8.6 shows an elastic collision where momentum is conserved.
An illustration shows before and after diagrams of two boxes moving toward each other on a frictionless surface. The box on the left is labeled m one and the box on the right is labeled m two. Both diagrams are labeled System of Interest. In the before diagram, a velocity vector, v one, points from m one toward m two. A second, shorter velocity vector, v two, points from the left side of m two toward m one. Two equations are shown: p one plus p one equals p total and net F equals zero. In the after diagram, both velocity vectors point away from each mass, m one and m two. The vector on the left is shorter than the one on the right. The equation p one prime plus p one prime equals p total is shown.
Figure 8.6 The diagram shows a one-dimensional elastic collision between two objects.
An animation of an elastic collision between balls can be seen by watching this video. It replicates the elastic collisions between balls of varying masses.
Perfectly elastic collisions can happen only with subatomic particles. Everyday observable examples of perfectly elastic collisions don’t exist—some kinetic energy is always lost, as it is converted into heat transfer due to friction. However, collisions between everyday objects are almost perfectly elastic when they occur with objects and surfaces that are nearly frictionless, such as with two steel blocks on ice.
Now, to solve problems involving one-dimensional elastic collisions between two objects, we can use the equation for conservation of momentum. First, the equation for conservation of momentum for two objects in a one-dimensional collision is
p 1 + p 2 = p ′ 1 + p ′ 2 ( F net =0) .
Substituting the definition of momentum p = mv for each initial and final momentum, we get
m 1 v 1 + m 2 v 2 = m 1 v ′ 1 + m 2 v ′ 2 ,
where the primes (') indicate values after the collision; In some texts, you may see i for initial (before collision) and f for final (after collision). The equation assumes that the mass of each object does not change during the collision