A ballistics pendulum is a device used to measure the speed of a fast moving projectile such as a bullet. A bullet of mass 5g is fired into a large block of wood of mass 1.5kg suspended from some light wires. The bullet is stopped by the block and the entire system swings through a vertical distance of 5cm. Find the initial speed of the bullet.
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initial momentum = .005 v
momentum after impact = .005 v = 1.5 V
so
V = .005 v/1.5 = .00333 v
kinetic energy after impact = (1/2)(1.5)V^2 = 8.32*10^-6 v^2
final potential energy = mgh=1.5*9.81*.05
= 8.32*10^-6 v^2
so
v^2 = 88467
v=297 m/s
To find the initial speed of the bullet, we can use the principle of conservation of momentum in elastic collisions. The bullet and block system can be considered as a closed system.
Step 1: Determine the final velocity of the bullet and block system
Since the bullet gets embedded in the block and both move together after the collision, the final velocity of the bullet and block system can be determined using conservation of momentum.
The initial momentum of the system is zero since the bullet is at rest. Let's assume the final velocity of the bullet and block system is v.
Initial momentum = 0
Final momentum = (mass of bullet + mass of block) * final velocity
= (0.005 kg + 1.5 kg) * v
= 1.505 kg * v
Step 2: Determine the change in potential energy of the system
The entire system swings through a vertical distance of 5 cm, which we can convert to meters (0.05 m). This distance represents the change in potential energy of the system.
Change in potential energy = mass of system * acceleration due to gravity * change in height
The mass of the system is the sum of the mass of the bullet and the mass of the block (0.005 kg + 1.5 kg = 1.505 kg). The acceleration due to gravity is approximately 9.8 m/s^2.
Change in potential energy = 1.505 kg * 9.8 m/s^2 * 0.05 m
= 0.73835 J (Joules)
Step 3: Apply the conservation of mechanical energy
Since the system undergoes an elastic collision, the mechanical energy (the sum of kinetic and potential energy) is conserved.
Initial mechanical energy = Final mechanical energy
The initial mechanical energy consists of the initial kinetic energy of the bullet, while the final mechanical energy consists of the final kinetic energy of the bullet and block system, as well as the change in potential energy.
Initial kinetic energy = 0.5 * mass of bullet * initial velocity^2
Final kinetic energy = 0.5 * (mass of bullet + mass of block) * final velocity^2
0.5 * 0.005 kg * initial velocity^2 = (0.5 * 1.505 kg * v^2) + 0.73835 J
Simplifying the equation, we get:
0.0025 kg * initial velocity^2 = 0.7525 kg * v^2 + 0.73835 J
Step 4: Calculate the initial speed of the bullet
From step 3, we have the equation:
0.0025 kg * initial velocity^2 = 0.7525 kg * v^2 + 0.73835 J
We also know that the final velocity (v) is equal to the initial velocity (since the bullet gets embedded in the block). Therefore:
initial velocity = v
Substituting v for initial velocity in the equation, we get:
0.0025 kg * v^2 = 0.7525 kg * v^2 + 0.73835 J
Simplifying and rearranging the equation, we can solve for v:
0.7525 kg * v^2 - 0.0025 kg * v^2 = 0.73835 J
0.75 kg * v^2 = 0.73835 J
v^2 = 0.73835 J / 0.75 kg
v^2 = 0.98447 J/kg
v = sqrt(0.98447 J/kg)
v ≈ 0.992 m/s
Therefore, the initial speed of the bullet is approximately 0.99 m/s (rounded to two decimal places).
To solve this problem, we can use the principles of conservation of momentum and conservation of mechanical energy.
1. Conservation of momentum:
The momentum of the bullet before the collision is equal to the momentum of the bullet and block system after the collision, since there are no external forces acting on the system during the collision.
Let's denote the initial speed of the bullet as v. The momentum of the bullet before the collision is given by p_bullet = m_bullet * v, where m_bullet is the mass of the bullet.
2. Conservation of mechanical energy:
The mechanical energy of the system is conserved when there are no external forces doing work on the system. In this case, the only external force is due to friction, which is assumed to be negligible.
The initial mechanical energy of the system is given by E_initial = m_block * g * h, where m_block is the mass of the block, g is the acceleration due to gravity, and h is the vertical distance through which the system swings.
At the highest point in the swing, all of the initial mechanical energy is in the form of gravitational potential energy, so the final mechanical energy of the system is given by E_final = 0.
Since the bullet and block system moves through a vertical distance of 5 cm, equivalent to 0.05 m, we have:
E_initial = E_final
m_block * g * h = 0
1.5 kg * 9.8 m/s^2 * 0.05 m = 0
Now we can solve for the initial speed of the bullet.
From the conservation of momentum:
p_bullet = p_bullet+block
m_bullet * v = (m_bullet + m_block) * V
5 g * v = (5 g + 1.5 kg) * V
Dividing both sides of the equation by 5 g, we get:
v = (5 g + 1.5 kg) * V / (5 g)
Finally, substituting the known values:
v = (5 * 0.001 kg + 1.5 kg) * V / (5 * 0.001 kg)
Simplifying the equation, we find the initial speed of the bullet.