JEE Main 2019, 10 Jan Shift-I
LEVELJEE Advanced
Visualized Solution
The Sigma Insight: Head-on Collision
The Cinematic Setup
Imagine a scene straight out of an action movie: a tall building stands against the sky. From its very edge, a wooden block of mass is dropped, beginning its free fall under gravity. At that exact fraction of a second, a marksman on the ground fires a bullet of mass straight up. The bullet leaves the barrel at a blistering speed of .
They are on a direct collision course. Our mission is to find out exactly where they meet, what happens when the bullet embeds itself into the wood, and how high the resulting mangled mass will soar above the building's roof.
Phase 1
The Race to Collision
To find out when they collide, we could write out the position equations for both objects and set them equal. But there is a much more elegant tool in our physics arsenal: Relative Motion.
Since both the block and the bullet are in free fall (one moving down, one moving up), they both experience the exact same acceleration due to gravity, , acting downwards.
If we sit on the falling block and look at the bullet, the relative acceleration is zero!
This means, from the perspective of the block, the bullet is approaching at a constant relative velocity.
The initial distance separating them is the height of the building, . Since the relative speed is constant, the time it takes for them to meet is simply distance divided by speed:
They collide exactly after being released.
Phase 2
The Moment of Impact
Now that we know when they collide, we need to know where and how fast they are moving right before the crash. Let's analyze the block's journey during that .
Using the second equation of motion for the block dropping from rest:
The collision happens below the top of the building.
Next, let's find their individual velocities just a millisecond before impact.
For the block (falling down):
For the bullet (rising up):
Phase 3
The Perfectly Inelastic Embrace
The problem states the bullet gets embedded in the wood. This is the hallmark of a perfectly inelastic collision. During the incredibly short duration of the impact, the internal impulsive forces between the bullet and the wood are astronomically larger than the external force of gravity. Therefore, we can safely conserve linear momentum.
Let's establish a strict sign convention: Up is positive, Down is negative.
Initial Momentum = Final Momentum
Plugging in our values with their correct signs:
The result is positive! This means the bullet's upward momentum was so dominant that it completely reversed the block's downward motion. The combined mass now shoots upwards at .
Phase 4
The Final Ascent
We now have a single projectile of mass launched upwards from the collision point with an initial velocity of . How high will it go from this point?
We use the classic kinematic relation for maximum height:
The combined mass rises from the point of impact.
But the question asks for the maximum height reached above the top of the building.
Remember, the collision occurred below the top of the building. Therefore, the mass has to travel just to get back to the roof level. The remaining distance is how high it goes above the roof.
And there we have it! A beautiful interplay of relative kinematics, momentum conservation, and projectile motion leading us to the final answer of .
Similar Questions
JEE Main 2020, 8 Jan Shift-II
LEVELJEE Advanced
A particle of mass is dropped from a height above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of . If they collide head-on completely inelastically, then the time taken for the combined mass to reach the ground, in units of is
(A)
(B)
(C)
(D)
JEE Main 2021, 27 July Shift-I
LEVELJEE Main
Three objects and are kept in a straight line on a frictionless horizontal surface. The masses of and are and , respectively. moves towards with a speed of and makes an elastic collision with it. There after makes a completely inelastic collision with . All motions occur along same straight line. The final speed of is
(A)
(B)
(C)
(D)
LEVELJEE Main
A block of mass is moving with a speed of on a smooth surface. It strikes another mass of and then they move together as a single body. The energy loss during the collision is
(A)
(B)
(C)
(D)
JEE Advanced 1999
LEVELJEE Advanced
Two blocks of masses and are at rest on an inclined plane and are separated by a distance of as shown. The coefficient of friction between each block and the inclined plane is . The block is given a velocity of up the inclined plane. It collides with , comes back and has a velocity of when it reaches its initial position. The other block after the collision moves up and comes to rest. Calculate the coefficient of restitution between the blocks and the mass of the block . [Take and ]
JEE Main 2021, 17 March Shift-II
LEVELJEE Advanced
A rubber ball is released from a height of above the floor. It bounces back repeatedly, always rising to of the height through which it falls. Find the average speed of the ball. (Take, )
(A)
(B)
(C)
(D)
JEE Advanced 1993
LEVELJEE Advanced
Two blocks and each of mass , are connected by a massless spring of natural length and spring constant . The blocks are initially resting on a smooth horizontal floor with the spring at its natural length, as shown in figure. A third identical block , also of mass , moves on the floor with a speed along the line joining and , and collides elastically with . Then
* Multiple Correct Options
(A)
the kinetic energy of the - system, at maximum compression of the spring, is zero
(B)
the kinetic energy of the - system, at maximum compression of the spring, is
(C)
the maximum compression of the spring is
(D)
the maximum compression of the spring is
JEE Main 2019, 12 Jan Shift-I
LEVELJEE Advanced
A simple pendulum is made of a string of length and a bob of mass , is released from a small angle . It strikes a block of mass , kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle . Then, is given by
(A)
(B)
(C)
(D)
JEE Main 2021, 25 July Shift-1
LEVELJEE Advanced
A body of mass moving with a speed of makes an elastic collision with another body at rest and continues to move in the original direction but with one-fourth of its initial speed. The speed of the two body centre of mass is . Then, the value of is ......... .
JEE Main 2019, 9 Jan Shift-I
LEVELJEE Main
Three blocks and are lying on a smooth horizontal surface as shown in the figure. and have equal masses while has mass . Block is given an initial speed towards due to which it collides with perfectly inelastically. The combined mass collides with , also perfectly inelastically th of the initial kinetic energy is lost in whole process. What is value of ?
(A)
4
(B)
2
(C)
3
(D)
5
JEE Main 2019, 9 April Shift-I
LEVELJEE Main
A body of mass 2 kg makes an elastic collision with a second body at rest and continues to move in the original direction but with one-fourth of its original speed. What is the mass of the second body?
(A)
1.5 kg
(B)
1.2 kg
(C)
1.8 kg
(D)
1.0 kg