A 7.0-g bullet is fired into a 1.5-kg ballistic pendulum. The bullet emerges from the block with a speed of 200 m/s, and the block rises to a maximum height of 12.3 cm. Find the initial speed of the bullet?
From the height H that the block rises you can deduce its velocity just after the bullet passes through, Vblock'. You will have to ignore the mass loss in the bullet'ds opath, however. The apostrophe
" ' " denotes just after collision
Vblock' = sqrt(2 g H)
Vbullet' = 200 m/s
Mbullet*Vbullet = Mbullet*Vbullet' + Mblock*Bblock'
Solve for Vbullet
Well, let's solve this riddle, shall we? We have a bullet that goes whoosh into a pendulum and makes it go whee! Now, we need to find out the initial speed of the bullet.
First, let's find the change in kinetic energy of the bullet. The initial kinetic energy is given by 1/2mv^2, where m is the mass and v is the initial velocity. The final kinetic energy is given by 1/2mv'^2, where v' is the final velocity.
Since the bullet emerges with a speed of 200 m/s, the final kinetic energy is 1/2(7.0 g) (200 m/s)^2.
Now, let's find the change in potential energy of the pendulum. The mass of the pendulum is 1.5 kg and it rises to a maximum height of 12.3 cm. The change in potential energy is given by mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
Now, here comes the punchline. The change in kinetic energy of the bullet is equal to the change in potential energy of the pendulum. So, 1/2(7.0 g) (200 m/s)^2 = (1.5 kg)(9.8 m/s^2)(0.123 m). Now, let's solve this equation and see what we get.
And the answer is... drumroll please... the initial velocity of the bullet is approximately 238.3 m/s. Voila!
To find the initial speed of the bullet, we can use the principle of conservation of momentum.
1. First, let's calculate the momentum of the bullet after it emerges from the block. The momentum (p) of an object is calculated by multiplying its mass (m) by its velocity (v).
Momentum of bullet after emerging from the block = (mass of bullet) x (velocity of bullet)
Momentum of bullet after emerging = 7.0 g x 200 m/s
Note: The mass of the bullet needs to be converted from grams to kilograms. 1 kg = 1000 g.
Momentum of bullet after emerging = (7.0 g / 1000) kg x 200 m/s
2. The momentum of the bullet before hitting the block should be equal to the momentum of the bullet after emerging. In other words, the total momentum before and after the collision is conserved.
Momentum of bullet before hitting the block = (mass of bullet) x (initial velocity of bullet)
Since the bullet does not change its mass during the collision, we have:
Momentum of bullet before hitting the block = (7.0 g / 1000) kg x (initial velocity of bullet)
3. Since the bullet emerges with a speed of 200 m/s, we can replace the velocity in the momentum equation.
(7.0 g / 1000) kg x (initial velocity of bullet) = (7.0 g / 1000) kg x 200 m/s
Simplifying the equation:
initial velocity of bullet = 200 m/s
Therefore, the initial speed of the bullet is 200 m/s.
To find the initial speed of the bullet, we can use the principle of conservation of momentum and conservation of energy.
Step 1: Conservation of Momentum
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. The momentum of an object is given by the product of its mass and velocity.
The total momentum before the collision can be calculated as:
Initial momentum of the bullet = m_b * v_b, where m_b is the mass of the bullet and v_b is its initial velocity.
Initial momentum of the pendulum = m_p * 0, since the pendulum is initially at rest (velocity is zero).
The total momentum after the collision can be calculated as:
Final momentum of the bullet = m_b * v'_b, where v'_b is the velocity of the bullet after it emerges from the block.
Final momentum of the pendulum = m_p * v_p, where m_p is the mass of the pendulum and v_p is its final velocity.
Setting the initial and final momenta equal, we have:
m_b * v_b = m_b * v'_b + m_p * v_p
Step 2: Conservation of Energy
According to the principle of conservation of energy, the total mechanical energy before the collision is equal to the total mechanical energy after the collision. The mechanical energy of an object is given by the sum of its kinetic energy and potential energy.
The total mechanical energy before the collision can be calculated as:
Initial kinetic energy of the bullet = (1/2) * m_b * v_b^2
Initial potential energy of the pendulum = 0, since the pendulum is initially at the lowest point.
The total mechanical energy after the collision can be calculated as:
Final kinetic energy of the bullet = (1/2) * m_b * v'_b^2
Final potential energy of the pendulum = m_p * g * h, where g is the acceleration due to gravity and h is the maximum height reached by the block.
Setting the initial and final mechanical energies equal, we have:
(1/2) * m_b * v_b^2 = (1/2) * m_b * v'_b^2 + m_p * g * h
Now, we have two equations (momentum equation and energy equation) with two unknowns (v'_b and v_p).
Solving these equations simultaneously will give us the values of v'_b (velocity of the bullet after it emerges from the block) and v_p (velocity of the pendulum). Once we have these values, we can determine the initial velocity of the bullet (v_b) using the momentum equation.