To find the initial speed of the dart, we need to use the principle of conservation of momentum. This principle states that in an isolated system, the total momentum of the system remains constant. In this case, the system consists of the dart and the wood block, so the total momentum of the system before and after the collision must be the same.
We can find the momentum of the dart by using the formula p = mv, where p is the momentum, m is the mass, and v is the velocity. Before the collision, the dart has a mass of 5.1 g and is at rest, so its momentum is 0. After the collision, the dart has a velocity of 2.1 m/s and a mass of 5.1 g, so its momentum is 10.7 kg*m/s.
We can also find the momentum of the wood block by using the same formula. Before the collision, the wood block has a mass of 27.5 g and is at rest, so its momentum is 0. After the collision, the wood block has a velocity of 2.1 m/s and a mass of 27.5 g, so its momentum is 57.9 kg*m/s.
Since the total momentum of the system must remain constant, the momentum of the dart and the wood block after the collision must add up to 0. We can set up the following equation to solve for the initial velocity of the dart:
10.7 kgm/s + 57.9 kgm/s = 0
Solving for v, we find that the initial velocity of the dart is -5.37 m/s. This is the negative of the final velocity of the wood block, indicating that the dart and the wood block moved in opposite directions after the collision.
The final answer is 5.37 m/s.