The speed of ocean waves depends on their wavelength λ (in meters) and the gravitational field strength g (in m/sˆ2) in this way:

V=Kλˆp Gˆq

Where K is a dimentionless constant. Find the value of the exponents p and q.

Consider the dimensions. Left side, m/s.

Right side (dimentionless)(m)^p * (m/s^2)^q

m/s = m^P * m^q * 1/s^(2q)
consider the exponents of m
consider the exponents of s

You should be able to solve. Check my thinking and math.

Your thinking and math are correct.

Let's go through the steps together:

1. Start with the dimensional analysis. The left side of the equation is in meters per second (m/s).

2. On the right side, we have a dimensionless constant K, multiplied by wavelength raised to the power of p (in meters^p), and gravitational field strength raised to the power of q (in m/s^2)^q.

3. Apply the rules of dimensional analysis. On the right side, we multiply meters^p by meters^q, and divide by seconds^(2q).

4. Equate the exponents of meters on both sides. We have m^(1) on the left side (since the exponent is 1 in m/s), and m^(p+q) on the right side. Therefore, p+q = 1.

5. Equate the exponents of seconds on both sides. We have s^(-1) on the left side, and s^(-2q) on the right side. Therefore, -1 = -2q.

6. Now, we have a system of two equations: p+q = 1 and -1 = -2q.

7. Solve the second equation (-1 = -2q) for q. Divide by -2: q = 1/2.

8. Substitute the value of q into the first equation (p+q = 1): p + 1/2 = 1. Subtract 1/2 from both sides: p = 1 - 1/2 = 1/2.

Therefore, the exponents p and q are p = 1/2 and q = 1/2.