Optimizing Projectile Distance on an Inclined Plane

In summary, the conversation discusses the formula for determining the angle at which objects should be thrown on a hill to achieve the maximum distance. The two formulas mentioned are 1/2arctan(-cotϕ) = θ and π/4+ ϕ/2, with the latter being the more popular one. The conversation also mentions discrepancies between the two formulas at certain angles, and suggests searching for a derivation of the π/4+ ϕ/2 formula online.
  • #1
leitz
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Homework Statement


"A person stands at the base of a hill that is a straight incline making an angle ϕ with the horizontal. For a given initial speed v₀, at what angle θ (to the horizontal) should objects be thrown so that the distance d they land up the hill is as large as possible?"

I've looked up the same question on the internet, but the answers are either the formula (1/2arctan(-cotϕ) = θ) or π/4+ ϕ/2. I already figured out how to get 1/2arctan(-cotϕ) = θ, but I do not know how they got they π/4+ ϕ/2. Also, which is the correct answer?

Homework Equations



1/2arctan(-cotϕ) = θ
 
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  • #2
The π/4 + φ/2 formula predicts 45 degrees when φ=0, which is certainly correct. The arctan formula doesn't work at φ=0. At φ=45 degrees, the first formula predicts a reasonable 67.5 degrees, while the arctan formula seems to say -22.5 degrees. Better check my calc; I make mistakes.

The second formula is pretty well known. No doubt you could find a derivation of it on the web. Try searching for "Projectile motion on an incline". Of course it is always better to figure it out yourself if you have the time.
 

FAQ: Optimizing Projectile Distance on an Inclined Plane

What is a projectile on an inclined plane?

A projectile on an inclined plane refers to an object that is launched or thrown at an angle on a sloped surface. This type of motion is a combination of both horizontal and vertical motion.

What factors affect the motion of a projectile on an inclined plane?

The motion of a projectile on an inclined plane is affected by the initial velocity, angle of launch, and the slope of the plane. Other factors such as air resistance and friction may also play a role.

How is the motion of a projectile on an inclined plane calculated?

The motion of a projectile on an inclined plane can be calculated using the equations of motion, which take into account the initial velocity, angle of launch, and acceleration due to gravity. These equations can be used to find values such as the time of flight, maximum height, and range of the projectile.

What is the trajectory of a projectile on an inclined plane?

The trajectory of a projectile on an inclined plane is a curved path, known as a parabola. The shape of the parabola is determined by the initial velocity and angle of launch, and it is affected by gravity and air resistance.

What are some real-life applications of projectile motion on inclined planes?

Projectile motion on inclined planes can be observed in various real-life situations, such as a skateboarder going down a ramp, a ball rolling down a hill, or a car driving up a curved road. This type of motion is also important in sports, such as skiing and long jump, as well as in fields like engineering and physics.

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