Question

3. The ballistic pendulum is a device used to measure the speed of a projectile, such as a bullet. The projectile, of mass m, is fired into a large block of mass M, which is suspended like a pendulum. As a result of the collision, the pendulum and projectile together swing up to a maximum height h. Determine the relationship between the initial horizontal speed of the projectile, v, and the maximum heighth

Answer 1

Solution -

• Applying the conservation of the momentum equation -

Initial momentum = final momentum

mv = ( M + m )V_f

V_f   = mv / ( M + m )

Now we got the final velocity of the Block + bullet system now this system will move with V_f   and then it reaches to some height (H) till whole of it's kinetic  energy is converted into potential energy .

• kinetic  energy = (1/2) x ( M + m ) x V_f² [ Note - Here the mass is ( M + m ) ]
• potential energy = ( M + m ) x g x H   

Applying the energy conservation principle = >

• kinetic  energy = potential energy

  (1/2) x ( M + m ) x V_f²   =   ( M + m ) x g x H

H = (1/2g) x V_f²

• Putting the value of V_f   = mv / ( M + m ) in above equation of ' H ' we gets -

H = (1/2g) x [mv / ( M + m ) ]²

H / V² = m² / [2g( M + m )²]

This is the required relation between the initial velocity and the height .