Final answer:
To solve the equation of motion for the metal block, we can integrate both sides of the equation.
Step-by-step explanation:
To solve the equation of motion for the metal block, we need to find the velocity function, v(t). The resistive force acting on the block is given by F = -cv^(3/2), where c is a constant. Using Newton's second law, F = ma, we can write the equation of motion as ma = -cv^(3/2). Since acceleration is the derivative of velocity with respect to time, we have m(dv/dt) = -cv^(3/2). Rearranging the equation, we get dv/v^(3/2) = -(c/m)dt.
Now, we can integrate both sides of the equation. Integrate the left side by using the substitution u = v^(1/2). This gives us the integral of u^-3/2 du, which equals -2u^-1/2. Integrate the right side of the equation by using the substitution v = -(c/m)t + C, where C is a constant of integration. This gives us the integral of -(c/m)dt, which equals -(c/m)t + C. Equating the two integrals, we have -2u^-1/2 = -(c/m)t + C.
Using the initial condition V(0) = V0, which means u(0) = V0^(1/2), we can solve for C. Plugging in u(0) = V0^(1/2) and t = 0, we get -2(V0^(1/2))^-1/2 = -C. Simplifying, we have C = 2V0^(1/2).
Now, we can substitute C and u into the equation -2u^-1/2 = -(c/m)t + C. This gives us -2v^(-1/2) = -(c/m)t + 2V0^(1/2). Solving for v, we rearrange the equation to v^(-1/2) = (c/m)t - V0^(1/2). Taking the reciprocal of both sides, we get v^(-1/2)^(-1) = 1/v = (1/(c/m)t - V0^(1/2)). Rearranging the equation again, we arrive at v = 1/(1/(c/m)t - V0^(1/2)).