Maximum range equation

In summary, a projectile is launched at 80m/s at an angle x to the horizontal off the top edge of an infinitely long hill with a 35 degree angle. The maximum horizontal displacement equation can be derived by considering the projectile's motion along the x and y-axis separately. By setting the projectile's y-position to 0 when it touches the hill, the time it takes to fall can be calculated and substituted into the x-position equation to find the range along the hill. Multiplying this range by cos35 will give the horizontal range. To find the angle of maximum range, careful calculations must be done.
  • #1
hadroneater
59
0

Homework Statement


A projectile is launched 80m/s into the air at angle x to the horizontal off the top edge of an infinitely long hill(inclined plane). The hill makes a 35 degrees angle to the horizontal
a) Derive an equation for the range(maximum horizontal displacement).
b) At what angle will the projectile have maximum range?


Homework Equations


I know the equation for range of an even surface and how to derive it but I don't think that will be of help here.


The Attempt at a Solution


Is this question even possible?
I'm tried to derive an equation but with no luck. That's the hard part. I think I just need to differentiate the equation to get the angle of maximum range.
 
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  • #2
You can start with the usual equations and the components of the initial velocity in θ.

The slope adds the additional relationship between the x and y as in

y/x = tan35
 
  • #3
U have several way to approach this question.

1)U may establish the equation representing the way the projectile moving. Establish the equation of the hill. Find the cross point.

2)(I recommend) U consider the line representing the hill the x axis(picture). Do some math and find the result.

attachment.php?attachmentid=19150&stc=1&d=1243665976.jpg


Consider the projectile moves along x and y-axis seperately:

x axis: x=Vo.cosa.t - (sin35.g.t^2)/2 (1)
y axis: y=Vo.sina.t - (cos35.g.t^2)/2 (2)

When the projectile touch the hill, y = 0
So t=0 (starting point) or t= 2.Vo.sina/(cos35.g) (the time it falls)

Replace t= 2.Vo.sina/(cos35.g) in the equation (1), and u will find the range along the hill (pretty complicated).

To convert the range along the hill to the horizontal range, just multiply the x with cos35.

Now u get the result, try to find at which angle the range is max.

Hard question it is, but do it carefully and u can solve it. Be happy :D
 

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1. What is the maximum range equation?

The maximum range equation is a mathematical formula used to calculate the maximum distance that an object can travel when launched at a specific angle and velocity, assuming there is no air resistance. It is commonly used in physics and engineering to analyze projectile motion.

2. How is the maximum range equation derived?

The maximum range equation is derived from the basic principles of projectile motion, specifically the equations of motion for horizontal and vertical displacement. By setting the vertical displacement equal to zero and solving for the angle of launch, the maximum range can be determined.

3. What are the variables involved in the maximum range equation?

The variables involved in the maximum range equation are initial velocity, launch angle, acceleration due to gravity, and time. Other variables such as air resistance and wind speed can also be taken into account in more advanced versions of the equation.

4. Can the maximum range equation be used for any object?

No, the maximum range equation is specifically designed for objects that follow a parabolic trajectory, such as projectiles. It does not take into account other factors such as rotation or irregular shape, which can affect the range of an object.

5. How is the maximum range equation used in real-world applications?

The maximum range equation is commonly used in fields such as ballistics, aerospace engineering, and sports science to predict the distance that a projectile or object will travel. It is also used in designing and testing weapons, missiles, and other projectiles.

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