a 24 gramme bullet travelling at 198m/s buries itself in a 6.5kg pendulum hanging on a 2.2 m length of string, which makes the pendulum swing in an arc. determine the horizontal component of the displacement of the pendulum.
We need the velocity of the bullet-block system, so by conservation of momentum
m1v1 = (m1+m2)vf
solve for vf
Now we do conservation of energy:
1/2(m1+m2)vf^2 = (m1+m2)gh
solve for h
and don't forget to convert gms to kgs
To determine the horizontal component of the displacement of the pendulum, we need to analyze the conservation of linear momentum.
First, let's calculate the initial momentum of the bullet before it hits the pendulum. The formula for momentum is:
momentum = mass × velocity
The mass of the bullet is given as 24 grams, which is the same as 0.024 kg. The velocity of the bullet is given as 198 m/s. Therefore, the initial momentum of the bullet is:
momentum_bullet = (mass_bullet) × (velocity_bullet)
= 0.024 kg × 198 m/s
= 4.752 kg·m/s
According to the law of conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision. After the bullet buries itself in the pendulum, the pendulum and the bullet move together.
The pendulum consists of the bullet, which has a mass of 0.024 kg, and the pendulum itself, which has a mass of 6.5 kg. The total mass after the collision is therefore:
mass_total = mass_pendulum + mass_bullet
= 6.5 kg + 0.024 kg
= 6.524 kg
Now, let's calculate the velocity of the pendulum and bullet after the collision. To find the velocity, we use the equation:
momentum = mass × velocity
Since the total momentum is conserved, we can solve for the velocity after the collision:
momentum_total = mass_total × velocity_total_after
Substituting the values we have:
4.752 kg·m/s = 6.524 kg × velocity_total_after
Solving for velocity_total_after:
velocity_total_after = 4.752 kg·m/s / 6.524 kg
= 0.727 m/s
The velocity we calculated is the total velocity, including both the vertical and horizontal components. To find the horizontal component of the displacement, we only need to consider the horizontal part of the velocity. Since the vertical component does not affect the horizontal displacement, we can ignore it.
The horizontal displacement is equal to the distance the pendulum moves in the horizontal direction. Considering the length of the pendulum string is given as 2.2 m, the horizontal displacement is also 2.2 m.
Therefore, the horizontal component of the displacement of the pendulum is 2.2 m.