Question

Resistance Proportional to Velocity It is reasonable to assume that the air resistance encountered by a moving object, such as a car coasting to a stop, is proportional to the object's velocity. The resisting force on an object of mass \(m\) moving with velocity \(v\) is thus \(-k v\) for some positive constant \(k\). (a) Use the law Force = Mass \(\times\) Acceleration to show that the velocity of an object slowed by air resistance (and no other forces) satisfies the differential equation $$m \frac{d y}{d t}=-k v$$ (b) Solve the differential equation to show that \(v=v_{0} e^{-(k / m) t}\) where \(v_{0}\) is the velocity of the object at time \(t=0 . (c) If \)k$ is the same for two objects of different masses, which one will slow to half its starting velocity in the shortest time? Justify your answer.

Short answer

The velocity of an object slowed by air resistance satisfies the differential equation \(m \frac{dv}{dt} = -k v\). The velocity can be represented by the equation \(v = v_{0} e^{-\(\frac{k}{m}\) t}\), where \(v_{0}\) is the velocity at time \(t=0\). If two objects have the same value for \(k\) but different masses, the object with the smaller mass will slow to half its starting velocity in the shorter amount of time.

Step-by-step solution

Set up the differential equation

Starting with the law of motion - Force equals Mass times Acceleration or F = m*a. As acceleration is the first derivative of velocity, we can write it as a = dv/dt. The resisting force on an object of mass \(m\) moving with velocity \(v\) is \(-k v\). Therefore, \(-k v = m \frac{dv}{dt}\) is the differential equation representing the motion of the object.

Solve the differential equation

To solve the differential equation, separate the variables and integrate. \(-k v = m \frac{dv}{dt}\) becomes \(\frac{dv}{v} = -\frac{k}{m} dt\). Integration yields ln|v| = -\(\frac{k}{m}\)t + C. Exponentiating both sides to get rid of the natural logarithm, we get |v| = e^{-\(\frac{k}{m}\)t + C}, after some simplifications, we're left with \(v = v_{0} e^{-\(\frac{k}{m}\) t}\), where \(v_{0}\) is the velocity at time \(t=0\).

Compare objects of different mass

Considering two objects of different masses with the same \(k\), according to the velocity equation from step 2, the object with smaller mass will slow down to half its initial velocity quicker. This is because the rate factor \(\frac{k}{m}\) is inversely proportional to the mass. If the mass is less, the rate factor will be larger and thus it will take less time for the velocity to decrease to half of its original value.

Key concepts

Air Resistance

When objects move through the air, they experience a force commonly known as air resistance, which is essentially the friction between air molecules and the surface of the moving object. This force acts in the opposite direction to the motion of the object and increases with the object's speed. The effect can be observed in everyday life, such as when a leaf flutters down rather than falling straight or when a cyclist feels a stronger push against them as they speed up.

For many practical applications, the proportionality of air resistance and velocity provides a simplified model to describe this force. Mathematically, this relationship is given by the equation \( F_{\text{air}} = -k v \) where \( k \) is the positive constant of proportionality and \( v \) is the velocity of the object. This negative sign indicates that air resistance acts in the opposition to the direction of motion. As the velocity increases, the air resistance force increases linearly with \( v \) which is the reason why objects like cars require more force to accelerate at higher speeds. Understanding this concept helps explain a wide range of phenomena in physics and engineering, including how parachutes slow descent and how vehicles are designed to minimize drag for increased efficiency.

Velocity and Acceleration

Velocity is the rate of change in an object's position, typically measured in meters per second in the metric system. It is a vector quantity, meaning it has both magnitude and direction. Acceleration, on the other hand, is the rate of change of velocity. It also is a vector and is measured in meters per second squared \( (m/s^2) \). Whenever an object speeds up, slows down, or changes direction, it is accelerating.

The connection between velocity, acceleration, and force is described by Newton's second law of motion, which states that force equals mass times acceleration (\( F = m \times a \)). In our exercise, when considering the force of air resistance, the law is rearranged to express acceleration as the first derivative of velocity with respect to time (\( a = \frac{dv}{dt} \)). The air resistance force, being proportional to velocity as \(F_{\text{air}} = -k v\), generates an acceleration that affects the velocity of the object, leading to the differential equation \( m \frac{d v}{d t} = -k v \). Solving this equation provides a means to predict how an object's velocity will change over time under the influence of air resistance, which is essential for modeling motions in physics and various engineering applications.

Exponential Decay

Exponential decay is a process where the quantity decreases at a rate proportional to its current value. It occurs in various natural and technological processes, such as the cooling of a hot object in a room, the discharge of a capacitor in an electrical circuit, and the decay of radioactive substances.

In our context, the velocity of an object subjected to air resistance decreases exponentially over time. The term \(v_{0} e^{-\frac{k}{m} t}\) describes the velocity as a function of time, where \(v_{0}\) represents the initial velocity, \(e\) is the base of the natural logarithm, a mathematical constant approximately equal to 2.71828, \(k\) is the air resistance proportionality constant, and \(m\) is the mass of the object. The expression \(e^{-\frac{k}{m} t}\) is the factor that accounts for the exponential decrease in velocity. As time (\(t\)) progresses, the exponent becomes more negative, causing the value of \(e^{-\frac{k}{m} t}\) to shrink, leading to a decay in velocity towards zero. Understanding exponential decay not only aids in grasping the motion of objects with air resistance but also offers insights into a myriad of other phenomena in both physics and mathematics where similar decay patterns are observed.