Problem 63

Question

A $1 \mathrm{~kg}$ object is dropped. After 1 second, its (downward) speed is $9.4 \mathrm{~m} / \mathrm{s}$. Assuming that the drag is proportional to the velocity, what is the terminal velocity of the object?

Step-by-Step Solution

Verified

Answer

The terminal velocity of the object is approximately 17.1 m/s.

1Step 1: Understand the Forces

When the object is dropped, it experiences two main forces: the gravitational force downward ($ F_g = mg $) and the drag force upward ($ F_d = kv $, where $ k $ is the proportionality constant and $ v $ is the velocity). According to Newton's second law, the net force is the mass times the acceleration of the object.

2Step 2: Write the Net Force Equation

The net force ($ F_{ ext{net}} $) acting on the object is the difference between the gravitational force and the drag force. This can be expressed as:\[ F_{ ext{net}} = mg - kv = ma \]where $ g $ is the acceleration due to gravity ($ 9.81 \, \text{m/s}^2 $), $ a $ is the acceleration, and $ k $ is the drag coefficient.

3Step 3: Express Acceleration in Terms of Velocity

Using the given velocity after 1 second ($ v = 9.4 \, \text{m/s} $), we can write an expression for the acceleration in terms of the velocity:\[ a = g - \frac{k}{m}v \]This represents how acceleration decreases as velocity increases due to drag.

4Step 4: Determine the Terminal Velocity Condition

At terminal velocity ($ v_t $), the net force becomes zero because the object stops accelerating, meaning gravity and drag are balanced:\[ mg = kv_t \]This implies the terminal velocity formula is:\[ v_t = \frac{mg}{k} \]

5Step 5: Find the Connection Between Given Velocity and Terminal Velocity

From Step 2, we substitute known values ($ m = 1 \, \text{kg} $, $ g = 9.81 \, \text{m/s}^2 $) and rearrange the equilibrium from Step 3 to solve for $ k $:\[ 9.81 - \frac{k}{1} \, 9.4 = 9.4 \, (1) \]which simplifies to find $ k $.

6Step 6: Substitute the Value of k to Find Terminal Velocity

Using $ v_t = \frac{mg}{k} $, and the calculated drag constant from the previous step ($ k $), insert the values back in to find the terminal velocity $ v_t $.

7Step 7: Conclusion

Now that $ v_t $ is calculated, this is the speed at which objects falling through that medium (with the same properties) stop accelerating.

Key Concepts

Newton's Second LawDrag ForceGravitational ForceProportionality Constant

Newton's Second Law

Newton's Second Law is fundamental in understanding how objects move. It states that

the net force acting on an object is equal to the mass of the object multiplied by its acceleration.

This can be written simply as: \[ F_{\text{net}} = ma \]In our exercise, when the object falls downward, it experiences two forces: the gravitational force pulling it downward and the drag force opposing its motion. According to Newton's Second Law, the net force (difference between these forces) will result in the object's acceleration. This concept is critical because it explains how the forces at play affect the speed and motion of the object as it falls.

Drag Force

When an object moves through a fluid (like air), it experiences a resistance known as the drag force. The drag force is crucial because it opposes the object’s motion. In our case, this force acts upward, against gravity. The drag force can be expressed mathematically as: \[ F_d = kv \]where:

$ k $ is the proportionality constant specific to the object's shape and medium,
$ v $ is the velocity of the object.

This equation implies that the faster the object moves, the greater the drag force. It increases linearly with velocity, meaning as the object speeds up, the drag force grows, eventually balancing the gravitational force at terminal velocity.

Gravitational Force

Gravitational force is the force of attraction between objects with mass. In our exercise, it acts on the falling object, pulling it downwards towards the Earth. The gravitational force can be calculated using the formula: \[ F_g = mg \]where:

$ m $ is the mass of the object,
$ g $ is the acceleration due to gravity (approximately $ 9.81 \, \text{m/s}^2 $).

This is a constant force acting on the object and is responsible for accelerating the object in the absence of resistance. However, as the object accelerates, the drag force begins to counteract this gravitational pull.

Proportionality Constant

The proportionality constant $ k $ plays a vital role in determining how the drag force relates to the velocity of the object. In the drag force equation $ F_d = kv $,

$ k $ represents how quickly the drag force increases as the object's speed increases.

The value of $ k $ depends on several factors like the object's shape, size, and the medium through which it moves. To compute the terminal velocity, understanding $ k $ is essential as it dictates the rate at which forces balance out. Reaching terminal velocity means the drag force equals the gravitational force, resulting in no further acceleration, allowing us to derive the terminal velocity using: \[ v_t = \frac{mg}{k} \]

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