A ball moves in a fluid and experiences a deceleration (negative acceleration that reduces speed) due to the viscosity. For this particular fluid, acceleration is given by a = -kv3 where k = 0.0014 m-2s. If we start the clock when ball's velocity is 10 m/s, what will be the time in the clock (in seconds) when the ball’s velocity reduces to 5 m/s?

Question

BlackTom AI · Accepted Answer

To tackle the deceleration due to viscosity, we start from the given differential equation for velocity: dv/dt = -k v^3, where k = 0.0014 m^-2 s.

Next, separate variables to integrate: dv/v^3 = -k dt. This is the same as v^-3 dv on the left and -k dt on the right. Integrating both sides yields ∫ v^-3 dv = ∫ -k dt, which gives (-1/2) v^-2 = -k t + C.

Apply the initial condition: at t = 0, v = 10 m/s. Substituting, (-1/2)(1/100) = C, so C = -1/200 = -0.005. The velocity-time relationship becomes (-1/2)(1/v^2) = -k t - 0.005. If we multiply both sides by -2, we obtain 1/v^2 = 2k t + 0.01.

Now insert the target velocity v = 5 m/s and solve for t: 1/25 = 0.04 = 2k t + 0.01. Subtract 0.01 from both sides to get 0.03 = 2k t. With k = 0.0014, 2k = 0.0028, so t = 0.03 / 0.0028. This division yields t ≈ 10.714... seconds.

For practical reporting, you’d round to a reasonable precision based on the data given; here, 10.7 seconds is a typical presentation. The core steps rely on solving the separable differential equation for dv/dt = -k v^3 and applying the initial condition to determine the constant of integration, then evaluating at v = 5 m/s to obtain the elapsed time.

Note on the method: using separation of variables and recognizing that the integral of v^-3 is (-1/2) v^-2 is essential. An alternative but equivalent route is to rearrange the equation as d(v^-2)/dt = 2k, which leads to a linear relation in v^-2 and t, and yields the same result when you plug in the boundary condition and the target velocity.

Common pitfalls: forgetting the negative signs during separation, miscomputing the integral of v^-3, or mixing up the constants when solving for t. Carefully track the algebra and units throughout to avoid such mistakes.

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