a bullet of mass 0.015kg strikes a ballistic pendullum of mass 2kg.the centre of mass of the pendulum rises a verticle distance of 0.12m.assuming that the bullet remains embedded in the pendulum, calculate the bullets initial speed.(show all calculations and steps)
mv=(m+M)u
u =mv/(m+M),
(m+M)u²/2=(m+M)gh.
u=sqrt(2gh).
mv/(m+M)= sqrt(2gh)
v= (m+M)•sqrt(2gh)/m
To calculate the bullet's initial speed, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Before the collision:
The bullet's momentum, Mb = mass of the bullet * velocity of the bullet (unknown).
The pendulum's momentum, Mp = mass of the pendulum * velocity of the pendulum (initially at rest).
After the collision:
Since the bullet remains embedded in the pendulum, they move together as one system.
The combined momentum, M = (mass of the bullet + mass of the pendulum) * velocity of the system.
Since momentum is conserved, we can set up the equation:
Mb + Mp = M
Now, let's calculate the individual momenta and substitute them into the equation:
Bullet's momentum (Mb) = mass of the bullet * velocity of the bullet (unknown)
Pendulum's momentum (Mp) = mass of the pendulum * velocity of the pendulum (initially at rest)
Combined momentum (M) = (mass of the bullet + mass of the pendulum) * velocity of the system
Given information:
Mass of the bullet (mb) = 0.015 kg
Mass of the pendulum (mp) = 2 kg
Vertical distance the center of mass of the pendulum rises (h) = 0.12 m
From the problem statement, when the bullet strikes the pendulum, it rises a vertical distance of 0.12 m. This implies that the initial kinetic energy of the bullet converts into potential energy gained by the bullet-pendulum system.
With this information, we can determine the initial speed of the bullet.
Step 1: Find the velocity of the pendulum after the collision.
Using the principle of conservation of mechanical energy, we can equate the potential energy gained by the bullet-pendulum system to the initial kinetic energy of the bullet.
mg * h = (1/2) * (mb + mp) * v^2
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values:
(2 kg) * (9.8 m/s^2) * (0.12 m) = (1/2) * (0.015 kg + 2 kg) * v^2
This equation gives us the velocity (v) of the bullet-pendulum system.
Step 2: Find the initial velocity (v0) of the bullet.
Since the bullet is embedded in the pendulum, we can write the equation:
Mb = (mb + mp) * v0
Substituting the given values:
(0.015 kg) * v0 = (0.015 kg + 2 kg) * v
Now we can solve for v0, the initial velocity of the bullet.
v0 = (0.015 kg + 2 kg) * v / 0.015 kg
After calculating the previous steps, you will have the final value of the bullet's initial velocity (v0). Plug in the appropriate numbers and solve the equation to find the answer.