# Projectile Motion - Incline Plane

• Harmony
In summary, the conversation discusses a problem involving finding the optimal angle for throwing objects up a hill to achieve the maximum distance. The person is struggling to eliminate the variable t from the equations and is advised to use the parabolic trajectory of the objects as a starting point. They are also given equations for the horizontal and vertical components and are encouraged to manipulate them to find the final solution.
Harmony
A person stands at the base of a hill that is a straight incline making an angle x with the horizontal. For a given initial speed v, at what angle y (to the horizontal) should objects be thrown so that the distance d they land up the hill is as large as possible?

I realize that tan x = vertical displacement/horizontal displacement, and d^2=horizontal distance^2+vertical distance^2. But no matter how do I manipulate the equation, it seems that I can't get rid of the variable t. Any help would be aprreciated.

You don't say enough about what you have done for me to figure out where you are in solving this problem.

The objects will follow a parabolic trajectory (I'm sure you know this). So perhaps starting with that would be a good first step.

Dorothy

If you write your usual equations for your horizontal(dx) and vertical(dy) components in you should have something of the form;

$$d_{x}=v_{i}\cos(x)t$$

Do the same for your vertical component. You now have two functions of time. Also note that;

$$d_{x}=d\cos(y)$$

and

$$d_{y}=d\sin(y)$$

You should then equate the appropriate formulae and then manipulate your horizontal displacement into some of the form t=.... Can you go from here and figure out the last step yourself?

## 1. What is projectile motion on an incline plane?

Projectile motion on an incline plane refers to the movement of an object that is launched or thrown at an angle on a sloped surface. The object follows a curved path as it moves both horizontally and vertically due to the forces of gravity and the incline of the plane.

## 2. How is the angle of the incline plane related to the trajectory of the projectile?

The angle of the incline plane has a significant impact on the trajectory of the projectile. The steeper the incline, the greater the vertical component of the projectile's motion will be, resulting in a shorter distance traveled. On the other hand, a shallower incline will result in a longer horizontal distance traveled by the projectile.

## 3. What factors affect the range of the projectile on an incline plane?

The range of the projectile on an incline plane is affected by the initial velocity, angle of inclination, and the gravitational force. Additionally, air resistance and the friction between the projectile and the incline plane can also impact the range of the projectile.

## 4. How does the mass of the projectile affect its motion on an incline plane?

The mass of the projectile does not have a significant impact on its motion on an incline plane. As long as the initial velocity and angle of inclination are the same, the motion of the projectile will be the same regardless of its mass. However, a heavier projectile may experience slightly more air resistance and friction, affecting its range.

## 5. Can the projectile's motion on an incline plane be predicted using mathematical equations?

Yes, the motion of a projectile on an incline plane can be predicted using mathematical equations, such as the equations of motion and trigonometric functions. These equations take into account the initial velocity, angle of inclination, and other factors to determine the trajectory and range of the projectile.

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