When an object falls down it experiences air friction. Assume that the friction is proportional to the speed 
v
(
t
)
v(t) (this is only true when the speed is low). 



Using Newton's law 
F
=
m
a
=
m
d
v
d
t
F=ma=m 
dt
dv
​
  this gives the differential equation

     
m
d
v
d
t
=
m
g
−
k
v
m 
dt
dv
​
 =mg−kv.

with initial condition 
v
(
0
)
=
0
v(0)=0. Here 
F
F is the sum of the gravitational force 
m
g
mg, where 
g
g is the gravitational constant, and the friction force 
−
k
v
−kv.  

(Note 
v
v is positive when going down.)

Give a formula for the speed 
v
(
t
)
v(t) for the period during which the object falls.

 
 
 
 
 
 


We are tasked with solving the first-order linear differential equation:

\[
m \frac{dv}{dt} = mg - kv
\]

where:
- \( m \) is the mass of the object,
- \( g \) is the acceleration due to gravity,
- \( k \) is the proportionality constant of the air resistance (friction),
- \( v(t) \) is the velocity as a function of time.

Additionally, we have the initial condition: \( v(0) = 0 \).

### Step 1: Rearranging the Equation
First, we can rewrite the differential equation as:

\[
\frac{dv}{dt} = g - \frac{k}{m} v
\]

### Step 2: Solving the Differential Equation
This is a first-order linear differential equation. To solve it, we can use the method of integrating factors.

The equation can be written in the form:

\[
\frac{dv}{dt} + \frac{k}{m} v = g
\]

Now, the integrating factor \( \mu(t) \) is given by:

\[
\mu(t) = e^{\int \frac{k}{m} dt} = e^{\frac{k}{m}t}
\]

Multiplying both sides of the differential equation by \( \mu(t) \):

\[
e^{\frac{k}{m}t} \frac{dv}{dt} + e^{\frac{k}{m}t} \frac{k}{m} v = g e^{\frac{k}{m}t}
\]

This simplifies to:

\[
\frac{d}{dt} \left( e^{\frac{k}{m}t} v \right) = g e^{\frac{k}{m}t}
\]

### Step 3: Integrating Both Sides
Now, integrate both sides with respect to \( t \):

\[
e^{\frac{k}{m}t} v = \int g e^{\frac{k}{m}t} dt
\]

The right-hand side is:

\[
\int g e^{\frac{k}{m}t} dt = \frac{g m}{k} e^{\frac{k}{m}t}
\]

So, we have:

\[
e^{\frac{k}{m}t} v = \frac{g m}{k} e^{\frac{k}{m}t} + C
\]

where \( C \) is the constant of integration. Dividing through by \( e^{\frac{k}{m}t} \), we get:

\[
v = \frac{g m}{k} + C e^{-\frac{k}{m}t}
\]

### Step 4: Applying the Initial Condition
To determine \( C \), we use the initial condition \( v(0) = 0 \):

\[
0 = \frac{g m}{k} + C e^{0}
\]

This gives:

\[
0 = \frac{g m}{k} + C \quad \Rightarrow \quad C = -\frac{g m}{k}
\]

### Step 5: Final Formula for \( v(t) \)
Substitute the value of \( C \) into the general solution:

\[
v(t) = \frac{g m}{k} - \frac{g m}{k} e^{-\frac{k}{m}t}
\]

Factor out \( \frac{g m}{k} \):

\[
v(t) = \frac{g m}{k} \left( 1 - e^{-\frac{k}{m}t} \right)
\]

Thus, the formula for the velocity \( v(t) \) during the period of the object's fall is:

\[
v(t) = \frac{g m}{k} \left( 1 - e^{-\frac{k}{m}t} \right)
\]

### Interpretation:
- As \( t \to \infty \), \( v(t) \to \frac{g m}{k} \), which is the terminal velocity. 
- For small \( t \), the speed starts at 0 and gradually increases.

### Would you like details or have any questions on this? Here are 5 related questions:

1. How can you determine the terminal velocity of an object in free fall with air resistance?
2. What happens to the velocity if the friction constant \( k \) increases?
3. How would the solution change if the initial velocity wasn't zero?
4. Can you derive the time it takes to reach half of the terminal velocity?
5. How does the mass \( m \) of the object affect its velocity over time?

**Tip**: In many problems involving air resistance, terminal velocity is a key concept that helps determine when forces reach equilibrium.

When an object falls down it experiences air friction. Assume that the friction is proportional to the speed v(t) (this is only true when the speed is low). Using Newton's law F = ma = m dv/dt, this gives the differential equation mdv/dt = mg - kv. with initial condition v(0) = 0. Here F is the sum of the gravitational force mg, where g is the gravitational constant, and the friction force -kv. (Note v is positive when going down.) Give a formula for the speed v(t) for the period during which the object falls.

Learn how to solve a first-order linear differential equation modeling the velocity of an object falling with air resistance. This problem covers Newton's law of motion, the effects of air friction, and the concept of terminal velocity. Suitable for advanced high school or undergraduate students.

Math Problem Statement

Using Newton's law F

m a

m d v d t

with initial condition v ( 0 )

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation