A walker's speed, v, is proportional to the ratio of his leg length, L, and the period of the repeating motion of his legs, T, that is, v ∝ L/T. If the period is measured to be proportional to Lp, where p = 3/4, what power of L must the speed be proportional to? Enter your answer as a fraction (in lowest terms), first the numerator, then the denominator.
The power of L that the speed must be proportional to can be determined by combining the two proportional relationships.
We know that the period T ∝ L^p, where p = 3/4. Therefore, we can write T = kL^(3/4), where k is a constant of proportionality.
From the first proportional relationship, v ∝ L/T, we can substitute T with kL^(3/4), resulting in v ∝ L / (kL^(3/4)).
Simplifying the expression, we get v ∝ 1 / (kL^(-1/4)), which can be written as v ∝ L^(1/4) / k.
Thus, the speed must be proportional to L^(1/4). The power of L is therefore 1, and the power of the speed is 1/4.
To determine the power of L to which the speed is proportional, we can use the given relationship v ∝ L/T, where T is proportional to L^p.
Substituting T = k * L^p into the original equation, we get v = k * L / (k * L^p), where k is a constant.
Simplifying the equation, we have v = L^(1-p) / k.
From this equation, we can see that the speed v is proportional to L^(1-p), where (1-p) = (1 - 3/4) = 1/4.
Therefore, the power of L to which the speed is proportional is 1/4.
To determine the power of L to which the speed is proportional, we need to express the proportionality relationship between v and L/T in terms of p.
Given that T is proportional to Lp, we can write T = kLp, where k is a constant of proportionality.
Now, substituting this expression for T into the proportionality relationship v ∝ L/T, we have v ∝ L/(kLp).
Simplifying this expression, we get v ∝ 1/(kLp-1).
Comparing this with the general form x ∝ Lq, where x is the quantity we want to find the power of L for and q is the power we're looking for, we can determine that q = -p + 1.
In this case, p = 3/4, so substituting this value into the expression for q, we have q = -(3/4) + 1 = 1/4.
Therefore, the power of L to which the speed is proportional is 1/4.