A 6.2 g piece of clay is propelled horizontally onto a block of wood with a mass of 18.4 g. The wood block is initially at rest on a 1.75 m tall post. After the collision the wood block will land 2.2 m from the base of the post. Find the initial speed of the clay.

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

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In this case, the clay and the wood block can be considered as a closed system because no external forces are mentioned.

The momentum of an object is equal to its mass multiplied by its velocity. Mathematically, it can be represented as:

Momentum = mass x velocity

We can write the conservation of momentum equation as:

m1 * v1 + m2 * v2 = m1 * u1 + m2 * u2

Where:
m1 = mass of the clay (6.2 g)
v1 = velocity of the clay (initial velocity)
m2 = mass of the wood block (18.4 g)
v2 = velocity of the wood block (initially at rest)
u1 = velocity of the clay after the collision
u2 = velocity of the wood block after the collision

Let's calculate the velocities after the collision:

Since the post is 1.75 m tall, the time it takes for the wood block to fall from the top to the base can be calculated using the formula:

s = ut + (1/2) * g * t^2

Where:
s = distance (1.75 m)
u = initial velocity (0, as the wood block is initially at rest)
g = acceleration due to gravity (9.8 m/s^2)
t = time

Rearranging the formula, we get:

t = sqrt((2s) / g)

Substituting the values, we have:

t = sqrt((2 * 1.75) / 9.8) ≈ 0.581 s

Now, we can use this time to calculate the velocity of the wood block after the collision using the formula:

v = u + gt

Since the wood block falls vertically downward due to gravity,

v2 = 0 + 9.8 * 0.581 ≈ 5.686 m/s

Now, we can substitute the values into our conservation of momentum equation:

(6.2 g) * v1 + (18.4 g) * 0 = (6.2 g) * u1 + (18.4 g) * 5.686 m/s

Simplifying the equation:

(6.2 g) * v1 = (18.4 g) * 5.686 m/s

Dividing both sides by (6.2 g):

v1 = (18.4 g * 5.686 m/s) / (6.2 g)

v1 ≈ 16.91 m/s

Therefore, the initial speed of the clay is approximately 16.91 m/s.