Chapter 6.4, Problem 53E

(a)

To determine

To show: The velocity of an object which is slowed by an air resistance satisfies the differential equation.

Answer to Problem 53E

The velocity of an object satisfies the differential equation.

Explanation of Solution

Given:

Air resistance is proportional to object’s velocity. The mass of an object is m and the velocity is v.

Calculation:

Acceleration can be find out as: $\frac{dV}{dt}$

As, we know force is defined as:

  $F=ma$

Where:

F is the force.

m is the mass of an object.

a is the acceleration.

  $\begin{array}{l}F=m\frac{dV}{dt}\\ \frac{dV}{dt}=-kV\\ m\frac{dV}{dt}=-kV\end{array}$

Hence, velocity satisfies the differential equation.

Conclusion:

The velocity of an object satisfies the differential equation.

(b)

To determine

To show: $v=v_0{e}^{-(k/m)t}$

Answer to Problem 53E

The given equation is derived.

Explanation of Solution

Given:

Air resistance is proportional to object’s velocity. The mass of an object is m and the velocity is v.Calculation:

  $v=v_o{e}^{(\frac{-k}{m})t}$

Where: v_o is the velocity at time $t=0$ .

  $\begin{array}{l}m\frac{dV}{dt}=-kV\\ \frac{dV}{dt}=\frac{-kV}{m}\\ V=V_o{e}^{\left(-\frac{k}{m}\right)t}\end{array}$

Conclusion:

The given equation is derived.

(c)

To determine

The reason for use of two rules of thumb for computation.

Answer to Problem 53E

The object having greater mass will be slower to half its starting point in the shortest time.

Explanation of Solution

Given:

Air resistance is proportional to object’s velocity. The mass of an object is m and the velocity is v.

Calculation:

  $\frac{dV}{dt}=\frac{-kV}{m}$

If the k is same then, the object that has greater mass will be slower to half its starting point in the shortest time.

Conclusion:

The object having greater mass will be slower to half its starting point in the shortest time.