A 4.3 g dart is fired into a block of wood with a mass of 22.6 g. The wood block is initially at rest on a 1.4 m tall post. After the collision, the wood block and dart land 3.1 m from the base of the post. Find the initial speed of the dart.

To find the initial speed of the dart, we can use the principle of conservation of momentum.

The principle of conservation of momentum states that the total momentum of an isolated system remains constant before and after a collision. In this case, the dart and the wood block form an isolated system since no external forces are acting on them.

The momentum, denoted as p, is the product of an object's mass and its velocity. In equation form, it can be written as:

p = m * v

where p is the momentum, m is the mass, and v is the velocity.

Before the collision, the wood block is at rest, so its initial momentum is zero:

p_initial_block = m_block * v_block = 0

After the collision, both the wood block and the dart land at a distance of 3.1 m from the base of the post. We can use this information to calculate the final momentum of the system:

p_final_system = (m_block + m_dart) * v_final_system

Since the total momentum is conserved, we can equate the initial momentum to the final momentum:

p_initial_block = p_final_system

Since the mass of the block and the initial velocities are both zero, we can simplify the equation to:

0 = (m_block + m_dart) * v_final_system

Rearranging the equation, we can solve for the final velocity of the system:

v_final_system = 0 / (m_block + m_dart) = 0

Now, we can calculate the initial velocity of the dart. The initial velocity of the dart will be the same as the final velocity of the system since there are no external forces acting on it:

v_initial_dart = v_final_system = 0

Therefore, the initial speed of the dart is 0 m/s. This means that the dart was not moving before it collided with the block of wood.