A bullet with a mass of 0.01kg is fired horizontally into a block of wood hanging on a string. The bullet sticks in the wood and causes it to swing upward to a height of 0.1m If the mass of the wood block is 2kg, what is the initial speed of the bullet?

281.1 m/s

5245

To find the initial speed of the bullet, we can use the law of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event, provided there are no external forces acting on the system.

Before the bullet hits the wood block, the total momentum is 0 because the wood block is initially at rest. After the bullet sticks in the wood block, they both move together as a combined system.

Let's calculate the momentum before the collision and after the collision:

Before the collision:
The momentum of the bullet before the collision is given by:
m_bullet * v_bullet, where m_bullet is the mass of the bullet and v_bullet is the initial velocity of the bullet.

After the collision:
The momentum of the combined system after the collision is given by:
(m_bullet + m_wood) * v_combined, where m_wood is the mass of the wood block and v_combined is the final velocity of the combined system.

According to the law of conservation of momentum, the momentum before the collision is equal to the momentum after the collision:

m_bullet * v_bullet = (m_bullet + m_wood) * v_combined

Now we can plug in the given values:

0.01 kg * v_bullet = (0.01 kg + 2 kg) * v_combined

Simplifying the equation:

0.01 kg * v_bullet = 2.01 kg * v_combined

Dividing both sides by 0.01 kg:

v_bullet = 2.01 kg * v_combined / 0.01 kg

Now, we need to find the final velocity of the combined system. The bullet and wood block start at rest and reach a maximum height of 0.1 m. We can use the principle of conservation of mechanical energy to solve for the final velocity.

At the highest point of the swing, all the initial kinetic energy is converted to potential energy. The potential energy of the system is given by:

PE = m_total * g * h

where m_total is the combined mass of the bullet and wood block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the maximum height reached by the system.

Setting the potential energy equal to the initial kinetic energy:

PE = KE_initial

m_total * g * h = 0.5 * m_total * v_combined^2

Let's plug in the given values:

(0.01 kg + 2 kg) * 9.8 m/s^2 * 0.1 m = 0.5 * (0.01 kg + 2 kg) * v_combined^2

Simplifying the equation:

2.01 kg * 9.8 m/s^2 * 0.1 m = 0.5 * 2.01 kg * v_combined^2

Dividing both sides by 0.5 * 2.01 kg:

9.8 m/s^2 * 0.1 m = v_combined^2

Simplifying further:

0.98 m/s^2 = v_combined^2

Taking the square root of both sides:

v_combined = sqrt(0.98 m/s^2)

v_combined ≈ 0.99 m/s

Now we can substitute the value of v_combined back into the equation for v_bullet:

v_bullet = 2.01 kg * 0.99 m/s / 0.01 kg

Simplifying the equation:

v_bullet = 2.01 kg * 0.99 m/s / 0.01 kg

v_bullet ≈ 198 m/s

Therefore, the initial speed of the bullet is approximately 198 m/s.