Projectile Motion and Explosion

< b="" />Projectile Motion Analysis:< />

- Let's consider a projectile of mass 3m that is launched into the air.
- The projectile follows a curved trajectory due to the force of gravity acting on it.
- The path of the projectile can be divided into two main components: horizontal motion and vertical motion.

< b="" />Horizontal Motion:< />

- In the absence of any external forces, the horizontal motion of the projectile is independent of its vertical motion.
- Therefore, the horizontal velocity of the projectile remains constant throughout its flight.
- The range of the projectile, which is the horizontal distance covered, can be determined using the equation: Range = Horizontal Velocity x Time of Flight.

< b="" />Vertical Motion:< />

- The vertical motion of the projectile is influenced by the force of gravity.
- The projectile rises to its highest point and then falls back down to the ground.
- At the highest point of its path, the projectile explodes into three equal parts.

< b="" />Effect of the Explosion:< />

- One part of the projectile retraces its path, which means it moves in the opposite direction with the same velocity.
- Another part comes to rest, implying that its velocity becomes zero.
- The third part continues to move in a new trajectory determined by the explosion.

< b="" />Impact on Range:< />

- The part that retracts its path will not contribute to the range.
- The part that comes to rest will also not contribute to the range as its velocity is zero.
- Therefore, only the remaining part will contribute to the range.
- Since the range is determined by the horizontal motion, the remaining part will continue to move horizontally with its original velocity.
- Thus, the range of the projectile will remain the same, which is 100 meters, regardless of the explosion.

< b="" />Distance of the Third Part:< />

- The distance of the third part from the point of projection when it finally lands on the ground depends on its new trajectory.
- If the explosion does not alter the direction of the third part, it will continue to travel in the same horizontal direction.
- In this case, the distance of the third part from the point of projection when it lands will also be 100 meters, the same as the range of the projectile.
- However, if the explosion imparts a different direction to the third part, its distance from the point of projection when it lands will vary.
- Without further information about the explosion's effect on the third part's trajectory, we cannot determine the exact distance.

In conclusion, the range of the projectile remains the same regardless of the explosion. However, the distance of the third part from the point of projection when it lands on the ground will depend on the new trajectory imparted by the explosion.

There are 2 ways:-

1.

During explosion there is no external force. so center of mass should not be altered.

if mass doesn't explode. it lands at the distance of 100 unit( range of projectile) from origion.




after explosion the center of mass of all 3 pieces should be on distance of 100 unit.


now out of 3 part of mass


one stop at highest point i.e at 50 unit

second would land at 100 unit.

if third would land at 150 sonly then it's center of mass is found at 100.

so 150 is the answer.


2.

use conservation of momentum.

At the 50 unit horizontal distance mass explodes.

let the velocity of mass at that instant be 3V

now 1 particle stop there.

2nd traces the same path means it has same velocity = 3v

using conservation of momentum velocity of 3rd comes out 6v.

both 2nd and 3rd will fall at ground at same time.

and 3rds velocity is double of 2nds, so its displacement would also he twice.


as the 2nd one hose from 50 to 100 unit horizontal displacement.

definitely 3rd one will go to 150.