Ball thrown off the top of a hill, ball hits hill

In summary: From there, the equations of motion should be simplerIn summary, Alice and Bob are having a competition to see who can throw a baseball the farthest from the top of a hill, with Bob trying to find the optimal angle to throw the ball. The equations used to solve for this optimal angle involve manipulating the initial equations and using derivatives to maximize f(θ). However, it is important to take into account the slope of the hill and the components of gravity when solving for the optimal angle.
  • #1
Jazzman
13
0

Homework Statement


Alice and Bob are having a competition to see who can throw a baseball the farthest from the top of a hill which slopes downward uniformly at an angle ϕ. Alice plays softball, and Bob doesn't have a particularly good arm, so he knows that he's probably going to lose. Fortunately, Bob has taken mechanics, and he realizes that there is an optimal angle θ at which he should throw the ball so that it has greatest range; what is that optimal angle?

Assume that Bob's initial velocity is fixed and the hill is infinitely tall and infinitely long, so that the ball always hits the hill.

2. The attempt at a solution
You don't have to look at my attempt to solve the equation, because it is fairly long.

My initial equations were:

x = v0cos(θ)t
y = (v0cos(θ)t)/tan(ϕ) + v0sin(θ)t - .5gt2
tan(ϕ) = x/y

I solved for t in the x equation, and got:
t = x/(v0cos(θ)

Then I substituted this value for t into the equation for y:
y = (v0cos(θ)x)/tan(ϕ)v0cos(θ) + (v0sin(θ)x)/(v0cos(θ) - .5g(x/(v0cos(θ))2

Then I substituted this value for y into the equation for tan(ϕ).

tan(ϕ) = x/(v0cos(θ)x)/tan(ϕ)v0cos(θ) + (v0sin(θ)x)/(v0cos(θ) - .5g(x/(v0cos(θ))2

Then I solved for x. After some rigorous simplification, x came out to be:
x = (2v02(cos2(θ) + sin(θ)cos(θ)tan(ϕ))/(g)tan(ϕ)

After that, I made a function of θ.
f(θ) = cos2(θ) + (a)sin(θ)cos(θ), where a = tan(ϕ)

x = (2v02f(θ))/tan(ϕ)

x is maximized when f(θ) is maximized. So I tried to find the derivative of f(θ), set it equal to 0, and all that stuff, but I'm having trouble finding a value of θ that maximizes f(θ). Here is the derivative, just so you know:

f'(θ) = -2sin(θ)cos(θ) +2(a)cos2(θ) - a
 
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  • #2
Thanks in advance for your help!
 
  • #3
Misread the question. My post applied to the situation in which you are not on a hill. Reading is important. My best advice would be to rotate your axes so that the x-axis is parallel to the hill, then split gravity into x and y components.
 
Last edited:

1. How does the height of the hill affect the trajectory of the ball?

The height of the hill will affect the potential energy of the ball, which will in turn affect the speed and angle at which the ball is thrown off the hill. The higher the hill, the more potential energy the ball will have, resulting in a longer and higher trajectory.

2. What is the impact of the angle at which the ball is thrown off the hill?

The angle at which the ball is thrown off the hill will determine the initial velocity and direction of the ball. If the ball is thrown at a higher angle, it will have a shorter and higher trajectory. Conversely, if the ball is thrown at a lower angle, it will have a longer and lower trajectory.

3. What factors influence the distance the ball travels after hitting the hill?

The distance the ball travels after hitting the hill is influenced by several factors, including the initial velocity, angle of throw, air resistance, and the elasticity of the ball and the hill. These factors all play a role in determining the speed and direction of the ball after it hits the hill.

4. How does the weight of the ball affect its trajectory?

The weight of the ball will affect the force with which it is thrown off the hill. A heavier ball will require more force to be thrown at the same angle and velocity as a lighter ball, resulting in a different trajectory. However, the weight alone does not determine the trajectory, as other factors such as air resistance and elasticity also play a role.

5. Can the ball's trajectory be predicted accurately?

The trajectory of the ball can be predicted accurately using mathematical equations and principles of physics, as long as all relevant factors are taken into account. However, unpredictable factors such as wind or imperfections in the surface of the hill may affect the accuracy of the prediction.

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