# Projectile with a linear air resistance force question

• joker_900
In summary, the conversation discusses finding the maximum horizontal range for a projectile with linear air resistance force, with equations for the maximum range and an approximation method for solving these equations. The first part involves finding the equation given, while the second part involves using numerical or approximation methods to solve the equation. The equations given are x/v0x * (v0y + g/k) + (g/k^2)ln(1 - kx/v0x) = 0 and x = 2v0yv0x/g - 8v0yv0xk/3g^2.
joker_900

## Homework Statement

OK I hope someone will help me with this

For a projectile with a linear air resistance force Fs =−mkv, show that the maximum horizontal range is given by the equation

(v0 + g/k)(x/u0) + (g/k^2)ln(1 - kx/u0) = 0

where u0 = V costheta, v0 = V sintheta are the horizontal and vertical components of the initial velocity.

The above equation cannot be solved in closed form, either it is solved numerically (e.g. using MAPLE) or it may be approximated, assuming that the correction with k =/= 0 is small. For the latter method show that

xmax = 2v0u0/g - 8v0*v0*u0*k/3g^2

## Homework Equations

(v0 + g/k)(x/u0) + (g/k^2)ln(1 - kx/u0) = 0

## The Attempt at a Solution

I've done the first part (finding the equation given) but I really don't know what the second part means (the bit about approximation). I tried using taylor series to 2 terms but this just gives the first term not the second, so I'm stumped. Can anyone shed some light on this?

I'm going to assume that the equations given are:

$$\frac{x}{v_{0x}} \left ( v_{0y} + \frac{g}{k} \right ) + \frac{g}{k^2} {\ln \left ( 1 - \frac{kx}{v_{0x}} \right ) } = 0$$

and

$$x = \frac{2v_{0y}v_{0x}}{g} - \frac{8k v_{0y}v_{0x}}{3g^2}.$$

Besides that, I have absolutely no idea. >_<

Hello! Thank you for reaching out for help on this problem. Solving for the maximum horizontal range of a projectile with linear air resistance can be quite complex, but I will try my best to explain the solution.

First, let's review the equation given in the homework statement: (v0 + g/k)(x/u0) + (g/k^2)ln(1 - kx/u0) = 0. This equation represents the trajectory of the projectile, taking into account the initial velocity (v0), gravitational acceleration (g), and the linear air resistance force (Fs = −mkv). The first term, (v0 + g/k)(x/u0), represents the horizontal distance traveled by the projectile, while the second term, (g/k^2)ln(1 - kx/u0), represents the vertical distance traveled.

Now, to solve for the maximum horizontal range, we need to find the value of x that will make the equation equal to 0. However, as you have already discovered, this equation cannot be solved in closed form. Therefore, we will need to approximate the solution. One way to do this is by using Taylor series, as you have attempted. However, to get a more accurate result, we will need to use more terms in the Taylor series expansion.

The second part of the homework statement suggests an alternative method of approximation, which assumes that the correction with k ≠ 0 is small. This means that we can ignore the second term, (g/k^2)ln(1 - kx/u0), in the equation and only consider the first term, (v0 + g/k)(x/u0). This simplifies the equation to (v0 + g/k)(x/u0) = 0. Solving for x, we get x = -v0u0/g. Substituting this value into the original equation, we get (v0 + g/k)(-v0u0/gu0) + (g/k^2)ln(1 + v0/g) = 0. Simplifying this further, we get (v0 + g/k)(-v0/g) + (g/k^2)ln(1 + v0/g) = 0. This equation can be further simplified to -v0^2/g + (g/k^2)ln(1 + v0/g) = 0. This is the

## 1. What is a projectile with a linear air resistance force?

A projectile with a linear air resistance force is an object that is propelled through the air and experiences a force that opposes its motion, known as air resistance. This force is directly proportional to the velocity of the object and acts in the opposite direction of its motion.

## 2. How does air resistance affect the motion of a projectile?

Air resistance slows down the motion of a projectile as it moves through the air. This is because the force of air resistance acts in the opposite direction of the projectile's motion, reducing its velocity and therefore its overall distance traveled.

## 3. What is the difference between linear and non-linear air resistance?

Linear air resistance is directly proportional to the velocity of the object, while non-linear air resistance is not. Non-linear air resistance is often seen in objects that have irregular shapes or experience turbulent air flow, resulting in a more complex and unpredictable force.

## 4. How is air resistance calculated in a projectile motion problem?

The most common way to calculate air resistance in a projectile motion problem is to use the formula FR = bv, where FR is the force of air resistance, b is a constant that depends on the characteristics of the object and the air, and v is the velocity of the object. This formula assumes that the air resistance force is directly proportional to the velocity, making it a linear force.

## 5. Can air resistance ever be ignored in a projectile motion problem?

In most practical situations, air resistance cannot be ignored in a projectile motion problem. However, in some cases where the velocity is relatively low or the object is very aerodynamic, the effects of air resistance may be minimal and can be ignored for simplicity. It is always important to consider the specific circumstances of the problem when determining whether or not air resistance can be neglected.

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