In a ballistic pendulum a bullet of mass 10kg is fired horizontally with a speed into a large wooden stationary block of mass 2kg that is suspended vertically by two cords as shown with a figure( not drawn) . In very short time the bullet penetrates the pendulum and remains embedded. The entire system tries to swing through a maximum height of 10m.
Find the relation that gives the speed V and In terms of the height and then find its value.
To solve this problem, we can use the principle of conservation of mechanical energy. Here are the steps to find the relation between the speed of the bullet and the height of the pendulum:
Step 1: Determine the initial mechanical energy of the system.
In this case, the initial mechanical energy is the kinetic energy of the bullet before it collides with the pendulum. Since the bullet is fired horizontally, it has only kinetic energy and no potential energy initially.
The initial kinetic energy (KE_initial) is given by the formula: KE_initial = (1/2) * m_bullet * v^2
Step 2: Determine the final mechanical energy of the system.
After the bullet hits the pendulum and becomes embedded, the entire system (bullet + pendulum) swings to a maximum height of 10m. At this maximum height, the system has only potential energy and no kinetic energy.
The final potential energy (PE_final) is given by the formula: PE_final = (m_bullet + m_pendulum) * g * h_max
Step 3: Apply the principle of conservation of mechanical energy.
According to the principle of conservation of mechanical energy, the initial mechanical energy is equal to the final mechanical energy. So we have the equation:
KE_initial = PE_final
Substituting the values we have:
(1/2) * m_bullet * v^2 = (m_bullet + m_pendulum) * g * h_max
Now, we can solve this equation to find the relation between the speed (V) and the height (h_max), and then determine the value of V.
Let me calculate it for you.
To find the relation between the speed of the bullet and the maximum height reached by the pendulum, we can consider the conservation of mechanical energy.
The initial mechanical energy of the system (bullet + pendulum) is equal to the final mechanical energy when the pendulum reaches its maximum height.
The initial mechanical energy is the kinetic energy of the bullet just before impact, and the final mechanical energy is the potential energy of the pendulum at its maximum height.
The initial kinetic energy of the bullet is given by:
KE_initial = (1/2) * m_bullet * V^2
The final potential energy of the pendulum is given by:
PE_final = m_pendulum * g * h
where:
m_bullet = mass of the bullet = 10 kg
V = speed of the bullet
m_pendulum = mass of the pendulum = 2 kg
g = acceleration due to gravity = 9.8 m/s^2
h = maximum height reached by the pendulum = 10 m
Setting the initial kinetic energy equal to the final potential energy, we have:
(1/2) * m_bullet * V^2 = m_pendulum * g * h
Solving for V, we get:
V = √[(2 * m_pendulum * g * h) / m_bullet]
Substituting the given values:
V = √[(2 * 2 * 9.8 * 10) / 10]
V = √(39.2)
V ≈ 6.26 m/s
Therefore, the relation between the speed V and the maximum height reached by the pendulum is:
V = √[(2 * m_pendulum * g * h) / m_bullet]
And for the given values, V ≈ 6.26 m/s.
First, we need to apply the principle of conservation of momentum. Since the bullet is fired horizontally, the initial momentum of the system is simply the momentum of the bullet, which is given by:
p_initial = m_bullet * v
where m_bullet is the mass of the bullet and v is its velocity. Since the bullet remains embedded in the pendulum, the final momentum of the system is simply the momentum of the pendulum after it has reached its maximum height. At this point, the pendulum is momentarily stationary (i.e. its velocity is zero) and all of its energy is potential energy. Thus, the final momentum of the system is zero.
Applying the principle of conservation of momentum, we can equate the initial and final momenta:
m_bullet * v = 0
Solving for v, we get:
v = 0 / m_bullet = 0
This means that the speed of the bullet is zero, which is obviously not possible. Therefore, there must be some mistake in the problem statement or the values given. Please double-check the information and provide any additional details if possible.