Problem 67 - Physics for Scientists and Engineers with Modern Physics

Question

(II) An object moving vertically has \(\vec{\mathbf{v}}=\vec{\mathbf{v}}_{0}\) at \(t=0\) . Determine a formula for its velocity as a function of time assuming a resistive force \(F=-b v\) as well as gravity for two cases: \((a) \vec{\mathbf{v}}_{0}\) is downward and \((b) \vec{\mathbf{v}}_{0}\) is upward.

Step-by-Step Solution

Answer

Velocity is \( v(t) = \frac{mg}{b} + (v_0 - \frac{mg}{b})e^{-\frac{b}{m}t} \) for both cases.

Step 1: Set Up the Problem

Consider the object moving vertically with an initial velocity \( \vec{\mathbf{v}}_0 \) at \( t = 0 \). The resistive force acting on the object is \( F = -bv \), where \( b \) is a constant, and gravity acts on it with force equal to \( mg \). Newton's second law gives us \( F = ma \).

Step 2: Write the Equation of Motion

For the object moving vertically, the equation of motion incorporating both gravity and resistive force is given by: \[ ma = mg - bv. \] Since \( a = \frac{dv}{dt} \), substitute to get: \[ m \frac{dv}{dt} = mg - bv. \] Rearranging gives: \[ \frac{dv}{dt} = g - \frac{b}{m}v. \]

Step 3: Integrate the Differential Equation

Rearrange the differential equation for integration: \[ \frac{dv}{g - \frac{b}{m}v} = dt. \] Integrate both sides: \[ \int \frac{dv}{g - \frac{b}{m}v} = \int dt. \] The left side integrates using the method of integrating by parts or recognizing it as a standard integral form, resulting in: \[ -\frac{m}{b} \ln|g - \frac{b}{m}v| = t + C. \]

Step 4: Solve for Velocity, Case (a) Downward Motion

Set \( \vec{\mathbf{v}}_0 \) as downward, meaning initial velocity is negative: \( v(0) = -v_0 \). Substitute initial conditions into the integrated equation to find \( C \). Solve for velocity \( v(t) \): \[ v(t) = \frac{mg}{b} + (v_0 - \frac{mg}{b})e^{-\frac{b}{m}t}. \] This expresses velocity as a function of time for downward initial velocity.

Step 5: Solve for Velocity, Case (b) Upward Motion

Set \( \vec{\mathbf{v}}_0 \) as upward, meaning initial velocity is positive: \( v(0) = v_0 \). Substitute initial conditions into the integrated equation to find \( C \). Solve for velocity \( v(t) \): \[ v(t) = \frac{mg}{b} + (v_0 - \frac{mg}{b})e^{-\frac{b}{m}t}. \] This expresses velocity as a function of time for upward initial velocity.

Key Concepts

Resistive Force

In the context of dynamics, a resistive force is a force that opposes the motion of an object. When an object moves through a medium, like air or water, the medium exerts a force opposite to the direction of the object's motion. This force can be expressed mathematically as \( F = -bv \), where \( b \) is a proportionality constant related to the nature of the medium, and \( v \) is the velocity of the object.
This negative sign indicates that the resistive force acts in the opposite direction to the velocity.

• The constant \( b \) can depend on factors like the shape of the object and the properties of the medium.
• Typically, higher velocities result in greater resistive forces, slowing the object down more effectively.
• Resistive forces are crucial in understanding the deceleration of objects in real-world scenarios.

By analyzing these forces, we can better predict how an object's speed changes over time.

Differential Equations

Differential equations form the backbone of motion analysis in physics. They relate functions to their derivatives, describing how a quantity changes over time. In the given exercise, the relationship between acceleration, velocity, and resistive force leads to a first-order linear differential equation: \( \frac{dv}{dt} = g - \frac{b}{m}v \).
This equation shows how the velocity of the object changes over time due to gravity and resistive forces.

• \( g \) is the acceleration due to gravity.
• \( \frac{b}{m} \) represents how strongly the resistive force influences the velocity.
• Such equations require solving for velocity as a function of time, \( v(t) \).

Understanding and solving differential equations is essential in describing real-world dynamic systems accurately.

Vertical Motion

Vertical motion refers to the movement of objects along a vertical axis under the influence of forces like gravity and resistance. In this scenario, the forces include gravity, pulling objects downward, and a resistive force, acting against their motion. The initial velocity at \( t = 0 \) plays a crucial role in how the object behaves over time.

• If the initial velocity is downward, the resistive force slows the object more quickly.
• If the initial velocity is upward, the object decelerates until gravity and resistance bring it to a halt, before it accelerates downward.
• The outcome of the motion also depends on whether the motion starts with a speed less than or greater than the terminal velocity.

Analyzing forces in vertical motion allows us to predict how long it takes for the object to stop moving upward or reach a certain speed.

Integration Techniques

To solve the differential equations governing motion, we use integration techniques. These techniques allow us to find the velocity as a function of time, \( v(t) \), from the equation \( \frac{dv}{dt} = g - \frac{b}{m}v \). The integration process involves separating variables: \( \frac{dv}{g - \frac{b}{m}v} = dt \), and then integrating each side.
Integration results in the expression: \[-\frac{m}{b} \ln|g - \frac{b}{m}v| = t + C\]

• The left side requires recognition of a standard integral form or integration by parts.
• Constant \( C \) is determined using initial conditions, such as initial velocity.
• Expressing \( v(t) \) provides insights into how quickly the object slows down due to resistive forces.

Using these techniques is crucial in deriving exact solutions from differential equations, letting us uncover the effects of resistance and gravity on velocity over time.