The initial speed of the bullet was approximately 136.5 m/s.
In the given ballistic pendulum scenario, we can determine the initial speed of the bullet by applying the principles of conservation of energy and considering the energy loss due to air resistance and friction. Here's the step-by-step solution:
Step 1: Calculate the Initial Potential Energy (PE)
The initial potential energy of the pendulum is equal to the gravitational potential energy of the block at the highest point of its swing. This can be calculated using the formula:
PE = mgh
where:
PE is the potential energy (J)
m is the mass of the block (kg)
g is the acceleration due to gravity (9.81 m/s²)
h is the height of the block above its lowest point (m)
In this case, the height of the block above its lowest point is equal to the length of the string, which is 1.91 m. Therefore, the initial potential energy of the pendulum is:
PE = (2.3 kg) (9.81 m/s²) (1.91 m) = 43.5 J
Step 2: Calculate the Final Kinetic Energy (KE)
The final kinetic energy of the pendulum is equal to the total kinetic energy of the block and the bullet at the lowest point of their swing. This can be calculated using the formula:
KE = (1/2)mv²
where:
KE is the kinetic energy (J)
m is the mass of the object (kg)
v is the velocity of the object (m/s)
In this case, the mass of the object is the combined mass of the block and the bullet, which is 2.3 kg + 0.0098 kg = 2.3098 kg. The velocity of the object is equal to the maximum speed of the pendulum at the lowest point of its swing. This speed can be calculated using the formula:
v = √(2gh)
where:
v is the velocity of the object (m/s)
g is the acceleration due to gravity (9.81 m/s²)
h is the height of the object above its lowest point (m)
In this case, the height of the object above its lowest point is equal to the length of the string, which is 1.91 m. Therefore, the maximum speed of the pendulum at the lowest point of its swing is:
v = √(2(9.81 m/s²)(1.91 m)) ≈ 5.84 m/s
Therefore, the final kinetic energy of the pendulum is:
KE = (1/2)(2.3098 kg)(5.84 m/s)² = 68.7 J
Step 3: Apply the Law of Conservation of Energy with Energy Loss
The law of conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. In this case, the initial potential energy of the pendulum is converted to the final kinetic energy of the pendulum, considering energy loss due to air resistance and friction. Let's assume a coefficient of restitution (COR) of 0.8, indicating that 80% of the initial kinetic energy of the bullet is transferred to the pendulum, while the remaining 20% is lost to air resistance and friction.
Therefore, we can rewrite the equation for the conservation of energy as:
0.8(KE_bullet) = KE
where:
KE_bullet is the initial kinetic energy of the bullet
Substituting the value we calculated earlier for KE, we get:
0.8(KE_bullet) = 68.7 J
Solving for KE_bullet, we get:
KE_bullet = 85.9 J
Now, we can calculate the initial speed of the bullet using the formula:
KE = (1/2)mv²
where:
KE is the kinetic energy of the bullet (J)
m is the mass of the bullet
In this case, the mass of the bullet is 0.0098 kg. Substituting the value of KE_bullet, we get:
85.9 J = (1/2)(0.0098 kg)v²
Solving for v, we get:
v ≈ 136.5 m/s
Therefore, the initial speed of the bullet was approximately 136.5 m/s.