How does the running velocity of an animal on level ground and uphill depend on the size \(L\) of the animal?

Before we start, let's define the main parameters and variables involved in the problem: - \(v_0\): velocity on level ground - \(v_1\): velocity when running uphill - \(L\): size of the animal (length) We want to find the relationship between the running velocities \(v_0\) and \(v_1\) and the size \(L\) of the animal.

To understand how an animal's speed is related to its size, we can apply concepts from biomechanics and physics. From biomechanics, we know that the force exerted by an animal's muscles during movement is proportional to the cross-sectional area of its muscles acting on its limbs (\(A \sim L^2\)). From physics, we know that work done is equal to force times distance (\(W = Fd\)), and power is equal to work done over time (\(P = W/t\)).

Let's now derive the relationship between the velocity of an animal on level ground (\(v_0\)) and its size (\(L\)). For an animal moving on level ground, the work done against gravity is zero, so the power exerted is solely used to overcome the friction force. We can now write the equation for power as: \(P = Fv_0\) Using the relationship between force and the cross-sectional area of the muscles, we can write: \(P \propto L^2v_0\) Since the animal's mass is proportional to its volume (\(M \sim L^3\)), and the power required to maintain a specific velocity is proportional to the mass of the animal, we have: \(P \propto ML^3\) Combining both equations, we get: \(L^2v_0 \propto L^3\) Dividing both sides by \(L^2\) gives: \(v_0 \propto L\) This means that the velocity of an animal on level ground is directly proportional to its size.

Next, let's derive the relationship between the velocity of an animal running uphill (\(v_1\)) and its size (\(L\)). When an animal runs uphill, the power exerted is used both to overcome the friction force and to lift the body against gravity (\(P = F_1v_1 + Mgh\), where \(h\) is the height and \(g\) is the acceleration due to gravity). Using the relationship between force exerted and cross-sectional area of muscles, we write: \(P \propto L^2v_1 + L^3h\) Since the height is given by \(h=vt\), we have: \(P \propto L^2v_1 + L^3v_1t\) For animals of the same species, the ratio of power generated over the mass is roughly constant. Therefore, we have: \(L^2v_1 + L^3v_1t \propto L^3\) Dividing both sides by \(L^2\) gives: \(v_1 + Lv_1t \propto L\) As \(t\) is independent of an animal's size, we have: \(v_1 \propto L\) So we find that the velocity of an animal running uphill is also directly proportional to its size.