JEE Main 2021, 27 July Shift-I
LEVELJEE Main
Visualized Solution
The Sigma Insight: Head-on Collision
Analyzing the Setup
Imagine a perfectly smooth, frictionless horizontal surface. On this surface, we have three blocks lined up: A, B, and C. Block A is the lightest with a mass of , while blocks B and C are heavier, each having a mass of .
The action begins when block A is given a push, sending it sliding towards block B with an initial speed of . This sets the stage for a fascinating sequence of two very different types of collisions.
The Elastic Encounter
When block A strikes block B, they undergo a perfectly elastic collision. This means that not only is the total momentum of the system conserved, but the total kinetic energy is conserved as well.
Since there are no external horizontal forces acting on the blocks, we can write the conservation of linear momentum equation:
By dividing the entire equation by the common mass , we get a much simpler relation:
Because the collision is perfectly elastic, the coefficient of restitution is exactly . This principle tells us that the relative velocity at which the blocks separate after the collision is equal to the relative velocity at which they approached each other before the collision:
Rearranging this gives us our second crucial equation:
Now, we have a system of two linear equations. Our goal is to find , the velocity of block B after the impact. By adding equation (i) and equation (ii) together, the terms cancel out beautifully:
Substituting the known initial velocity , we can easily solve for :
So, after the first collision, block B is propelled forward with a speed of .
The Inelastic Embrace
The story doesn't end there. Block B, now moving at , is on a direct collision course with the stationary block C. However, this second collision is completely inelastic.
In a completely inelastic collision, the colliding bodies do not bounce off each other; instead, they stick together and move as a single, combined mass. While kinetic energy is lost in this process (transformed into heat and sound), the total linear momentum remains strictly conserved.
Let's apply the conservation of momentum to this second event. The initial momentum is just the momentum of block B, and the final momentum is the combined mass of B and C moving at a new final velocity, :
Simplifying the masses, we get:
Solving for the final velocity , we find:
And there we have it! After the sequence of collisions, the combined mass of blocks B and C glides along the frictionless surface at a final speed of .
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