A 20g rifle bullet traveling 200m/s buries itself in a 3.3kg pendulum hanging on a 3.0m long string, which makes the pendulum swing upward in an arc.
Determine the vertical and horizontal components of the pendulum's maximum displacement.
conservationof momentum
0.020*200=(3.300+0.020)v solve for v, velocity of the pendulum after impact.
Now conservation of energy
totalmass*g*height=1/2 m v^2
solve for height the pendulum swings.
Then horizontal comes from diagram
Costheta=(3.0-h)/3.0
solve for theta
horiz distance=3.0 Tan Theta
65
oil
To determine the vertical and horizontal components of the pendulum's maximum displacement, we need to analyze the motion of the bullet when it hits the pendulum.
Step 1: Find the momentum of the bullet before it hits the pendulum.
The momentum (p) is given by the mass (m) multiplied by the velocity (v). Since the bullet mass is 20g (0.02kg) and the velocity is 200m/s, the momentum of the bullet is:
p = m * v = 0.02kg * 200m/s = 4 kg·m/s
Step 2: Calculate the total momentum after the bullet embeds itself in the pendulum.
Since the bullet embeds itself in the pendulum, the final momentum of the system (bullet + pendulum) is equal to the initial momentum of the bullet.
Therefore, the total momentum after the collision is also 4 kg·m/s.
Step 3: Determine the maximum displacement of the pendulum.
The initial momentum of the pendulum at the highest point of its swing is zero because the velocity is zero.
As the bullet embeds itself in the pendulum, the total momentum remains constant. Therefore, the final momentum when the pendulum reaches its maximum height is also 4 kg·m/s.
At the highest point of its swing, the pendulum's motion can be considered as purely vertical. Hence, we can equate the initial vertical momentum to the final momentum to find the maximum displacement in the vertical direction.
Step 4: Calculate the vertical component of the pendulum's maximum displacement.
The vertical component of the pendulum's maximum displacement can be found using the principle of conservation of momentum.
Initial momentum of the pendulum = 0 kg·m/s (at the highest point of swing)
Final momentum of the pendulum = 4 kg·m/s
Since momentum (p) is the product of mass (m) and velocity (v), we can write:
0 = m * v_max
4kg·m/s = (m + 3.3kg) * 0
0 = m + 3.3kg
m = -3.3kg
The negative value of 'm' implies that the pendulum mass is in the opposite direction to the velocity vector. This indicates that the pendulum moved in the opposite (upward) direction to the bullet's motion.
Therefore, in the vertical direction, the maximum displacement of the pendulum is 3.3m upward from its original position.
Step 5: Calculate the horizontal component of the pendulum's maximum displacement.
The horizontal component of the pendulum's maximum displacement can be determined based on the conservation of kinetic energy.
Since the bullet embeds itself within the pendulum, the final kinetic energy is shared between both the pendulum and bullet. Since no external forces act horizontally after the bullet embeds itself, the total horizontal momentum remains constant.
Therefore, the horizontal component will be the same as the horizontal component of the bullet's initial velocity.
The initial horizontal momentum of the bullet is:
p_horizontal = m * v_horizontal = 0.02kg * 200m/s = 4 kg·m/s
Hence, the horizontal component of the pendulum's maximum displacement is 4 m horizontally from its original position.
In summary:
- The vertical component of the pendulum's maximum displacement is 3.3m upward.
- The horizontal component of the pendulum's maximum displacement is 4m horizontally.