Question

A 4.0-cm-diameter, 55 g ball is shot horizontally into a tank of 40°C honey. How long will it take for the horizontal speed to decrease to 10% of its initial value?

Accepted Answer

Step 1: Identify the forces acting on the ball as it moves through the honey. The primary force is the drag force, which depends on the viscosity of the honey and the velocity of the ball. The drag force can be expressed as $$ F_d = 6 \pi \eta r v $$, where $$ \eta  is $$the dynamic viscosity of the honey, $$ r  is $$the radius of the ball, and $$ v  is $$the velocity of the ball. Step 2: Relate the drag force to the deceleration of the ball using Newton's second law. The net force acting on the ball is $$ F_d = m a $$, where $$ m  is $$the mass of the ball and $$ a  is $$its acceleration. Since $$ a = \frac{dv}{dt} $$, substitute $$ F_d $$ into the equation to get $$ 6 \pi \eta r v = m \frac{dv}{dt} . $$Step 3: Rearrange the equation to isolate $$ \frac{dv}{v}  on $$one side and $$ dt  on $$the other side. This gives $$ \frac{dv}{v} = -\frac{6 \pi \eta r}{m} dt . $$Note the negative sign, which indicates that the velocity is decreasing over time. Step 4: Integrate both sides of the equation to find the time $$ t  it $$takes for the velocity to decrease to 10% of its initial value. The limits of integration for $$ v $$ are $$ v_0  ($$initial velocity) to $$ 0.1 v_0 $$, and for $$ t $$, the limits are $$ 0  to$$ $$ t . $$The integral becomes $$ \int_{v_0}^{0.1v_0} \frac{1}{v} dv = -\int_{0}^{t} \frac{6 \pi \eta r}{m} dt . $$Step 5: Solve the integral to find $$ t . $$The left-hand side evaluates to $$ \ln(0.1) - \ln(v_0) $$, and the right-hand side evaluates to $$ -\frac{6 \pi \eta r}{m} t . $$Rearrange to solve for $$ t $$: $$ t = \frac{-\ln(0.1)}{\frac{6 \pi \eta r}{m}} . $$Substitute the given values for $$ \eta  ($$viscosity of honey at 40°C), $$ r  ($$radius of the ball), and $$ m  ($$mass of the ball) to compute the time.

A 4.0-cm-diameter, 55 g ball is shot horizontally into a tank of 40°C honey. How long will it take for the horizontal speed to decrease to 10% of its initial value?

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Key Concepts

Drag Force

Viscosity

Kinematics