A bullet of mass 20gm penetrates a board of thickness 10cm and mass 1kg at rest. If the board is free to move,find the thickness to which the bullet will penetrate.
I used conservation of momentum to find the combined velocity of bullet and the board in terms of the initial velocity of the bullet but can't figure out what to do next.
Ok, you now have the velocity of both, so figure KE of both combined. Now compare that to initial KE of the bullet, and the difference is work in the board.
the depth is due to bullet speed being great than board speed, when they are equal, it stops penetration.
Massbullet*timepenetrating=mbullet*changevelocity+massboard*changevelocityboard.
now figure time penetrating. Now figure average velocity of the board/bullet combination (vf/2).
depth=time/avgvelocity
After finding the combined velocity of the bullet and the board using conservation of momentum, you can now use the principle of conservation of energy to find the thickness to which the bullet will penetrate.
Let's assume the initial velocity of the bullet is "u", the final velocity of the bullet and the board combined is "v", and the thickness the bullet penetrates is "d".
The bullet and the board initially have kinetic energy, which is given by:
Initial kinetic energy = (1/2) * mass of the bullet * initial velocity of the bullet^2
The bullet and the board, after penetrating the board, have kinetic energy along with potential energy due to the rise in height. The final kinetic energy is given by:
Final kinetic energy = (1/2) * (mass of the bullet + mass of the board) * final velocity of the bullet and the board combined^2
Additionally, the bullet and the board have potential energy because they have been lifted to a height "d". The potential energy is given by:
Potential energy = (mass of the bullet + mass of the board) * acceleration due to gravity * d
Since there is no external work being done on the system, conservation of energy states that the initial kinetic energy is equal to the final kinetic energy plus the potential energy. Therefore, we can write the equation as:
(1/2) * mass of the bullet * initial velocity of the bullet^2 = (1/2) * (mass of the bullet + mass of the board) * final velocity of the bullet and the board combined^2 + (mass of the bullet + mass of the board) * acceleration due to gravity * d
Now, substitute the value of the final velocity derived from the conservation of momentum equation into the equation above. Solve for "d" to find the thickness to which the bullet will penetrate.