Final answer:
To find the initial speed of the dart, one must use the projectile motion equations to calculate the combined speed of the dart and wood block just after the collision, and then apply the conservation of momentum to determine the initial speed of the dart.
Step-by-step explanation:
Finding the Initial Speed of the Dart
The problem presented is a classic physics problem involving projectile motion and conservation of momentum. To find the initial speed of the dart, we must consider two main phases of the motion: the horizontal collision between the dart and the block of wood and the subsequent projectile motion of the combined dart and wood block. The approach will be to calculate the speed just after the collision using the distance it lands from the post and then use conservation of momentum to find the initial speed of the dart.
Firstly, when the wood block and dart land 2.9 m from the base of the post, we can use the equations of projectile motion to find their speed just after the collision. Since they drop from a height of 1.4 m, we calculate the time (t) it takes to fall using the equation for free fall:
t = √(2h/g), where h is the height and g is the acceleration due to gravity.
Now, knowing the time, we can determine the horizontal speed (v) required to travel 2.9 m:
v = d/t, where d is the distance.
Secondly, with the speed just after the collision known, we'll use the conservation of momentum principle which states that the initial momentum of the dart equals the final combined momentum of the dart and the wood block. The equation to use is:
m1 * v1 = (m1 + m2) * v2, where m1 and m2 are the masses of the dart and block respectively, v1 is the initial speed of the dart, and v2 is the speed just after the collision.
Rearranging for v1 gives us the initial speed of the dart, which completes our calculation.