# Finding the angle for maximum range with a constant initial velocity

• Failshire
In summary, the goal of this conversation was to understand the concept of maximum range in projectile motion and how to use the range equation to determine the angle that gives the maximum range for a given initial velocity. The conversation also discussed the relevant equations and identities, and the use of a Wired article to aid in understanding the concept. Ultimately, the maximum value of the sine function was identified as 1, and the angle of 45 degrees was determined to give the maximum range.
Failshire

## Homework Statement

According to the range equation (see below), what angle would give teh maximum range for a given initial velocity? Mathematically explain your hypothesis.

## Homework Equations

The equation given in our book for range:
R=(V$$_{o}$$cos$$\Theta$$)*(2V$$_{o}$$sin$$\Theta$$)/g

The equation given for range with same initial height and final height
R=Vo^2 sin(2$$\Theta$$)/g

The relevant identity of Sin:
sin(2$$\Theta$$)=2sin$$\Theta$$2cos$$\Theta$$

## The Attempt at a Solution

This is not so much about math as it is about understanding the math, which is unfortunately my weakness. So, please excuse any confusion or ignorance, but this is my thought process so far along with a website that has pretty much explained my question but that I am unable to understand.

-The goal is to find the angle at which range is maximum, for the same velocity. Range is X-Xinitial. What I need to do is be able to make an equation that let's me figure out what angle will make range as big as it can be.

This is where I'm stuck. I found an awesome piece in Wired magazine that does this equation for me, but I'm completely at a loss to understand what its doing. The piece tells me that the biggest sin can be is 1, but why? How do I know that sin can never be bigger than 1? When I set sin to 1, and use the identity to solve for the angle theta, I know it will give me 45 degrees.

I suppose its not so much a homework question but a theory question. I have the answer to the homework, but I'm completely at a loss to understand why it works that way and would benefit from a deeper knowledge (because knowing is half the fun)

I hope this isn't too rambly. Theory homework is rough!

Failshire said:
This is where I'm stuck. I found an awesome piece in Wired magazine that does this equation for me, but I'm completely at a loss to understand what its doing. The piece tells me that the biggest sin can be is 1, but why? How do I know that sin can never be bigger than 1? When I set sin to 1, and use the identity to solve for the angle theta, I know it will give me 45 degrees.

Because the nature of sin2θ or sin(nθ) [n=any real number] is such that its max. value is 1 and its minimum value is -1.

You can see this if y=sin2θ, dy/dθ= 2cos2θ, d2y/dθ2=-4sin2θ

For a stationary point, dy/dθ = 0 or 2cos2θ=0. Meaning that θ=45° (in this case θ varies from 0° to 90°)

when θ=45°, d2y/dθ2 = -ve meaning that θ=45° makes 'y' maximum.

## 1. What is the formula for finding the angle for maximum range with a constant initial velocity?

The formula for finding the angle for maximum range with a constant initial velocity is θ = tan-1(g/2v2), where θ is the angle, g is the acceleration due to gravity, and v is the initial velocity.

## 2. How do I determine the optimal angle for maximum range?

The optimal angle for maximum range can be determined by using the formula θ = tan-1(g/2v2). Plug in the values for g and v, and then use a calculator to find the inverse tangent of the resulting value.

## 3. What is the significance of finding the angle for maximum range?

Finding the angle for maximum range is important because it allows us to launch a projectile at the optimal angle in order to achieve the longest possible distance. This can be useful in fields such as sports, engineering, and military operations.

## 4. Can the angle for maximum range be calculated for any initial velocity?

Yes, the formula for finding the angle for maximum range with a constant initial velocity can be applied to any initial velocity. However, keep in mind that the angle may change depending on the initial velocity, as well as other factors such as air resistance and wind.

## 5. How does air resistance affect the angle for maximum range?

Air resistance can significantly affect the angle for maximum range by slowing down the projectile and reducing its overall distance. This means that the optimal angle for maximum range may need to be adjusted in order to compensate for air resistance.

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