A bullet of mass 45 g is fired at a speed of 220 m/s into a 5.0 kg sandbag hanging from a string from the ceiling. The sandbag absorbs the bullet and begins to swing. To what maximum vertical height will it rise?
use conservation of momentum to find initial velocity for the sandbag. Then, calculate the KE that sandbag has. Set that to the PE it will gain, solve for height.
Is there no kinetic energy after Collison when the sand bag is swinging?
To determine the maximum vertical height the sandbag will rise, we can use the principle of conservation of energy.
The initial kinetic energy of the bullet is given by:
KE_initial = (1/2) * m_bullet * v_bullet^2
where m_bullet is the mass of the bullet (45 g = 0.045 kg) and v_bullet is its velocity (220 m/s).
The kinetic energy is converted into potential energy as the sandbag rises, which can be calculated as:
PE_final = m_sandbag * g * h_max
where m_sandbag is the mass of the sandbag (5.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h_max is the maximum vertical height.
Since energy is conserved, we can equate the initial kinetic energy with the final potential energy:
KE_initial = PE_final
(1/2) * m_bullet * v_bullet^2 = m_sandbag * g * h_max
Substituting the given values:
(1/2) * 0.045 kg * (220 m/s)^2 = 5.0 kg * 9.8 m/s^2 * h_max
Simplifying the equation:
h_max = [(1/2) * 0.045 kg * (220 m/s)^2] / [5.0 kg * 9.8 m/s^2]
Calculating h_max:
h_max = (0.025 * 48400) / 49 = 25.88 m
Therefore, the sandbag will rise to a maximum vertical height of approximately 25.88 meters.
To find the maximum vertical height the sandbag will rise, we can use the principle of conservation of energy. The initial kinetic energy of the bullet is transferred to the sandbag in the form of gravitational potential energy as it rises.
First, let's find the initial kinetic energy (KE) of the bullet:
KE = (1/2) * mass * velocity^2
= (1/2) * 0.045 kg * (220 m/s)^2
= 539.4 J
Since the bullet is embedded in the sandbag and they move as one system, the total energy is conserved. This total energy is converted into gravitational potential energy (GPE) at the maximum height:
GPE = m * g * h
Where:
m is the total mass of the system (bullet + sandbag) = 0.045 kg + 5.0 kg = 5.045 kg
g is the acceleration due to gravity = 9.8 m/s^2
h is the maximum height we want to find.
Since the system starts from rest at the maximum height, all the initial kinetic energy is converted into gravitational potential energy. Therefore, we can set KE = GPE:
539.4 J = 5.045 kg * 9.8 m/s^2 * h
Now, let's solve for h:
h = 539.4 J / (5.045 kg * 9.8 m/s^2)
h ≈ 10.85 meters
Therefore, the maximum vertical height the sandbag will rise is approximately 10.85 meters.