Finding the Minimum θ for Rock Thrown at Speed v_0

In summary, the minimum value of θ, in terms of h and v_0, can be found by differentiating the tangent of the equation v_x = v_0cosα and setting it equal to zero. This will eliminate α and result in the equation v_0^2sin^2α = 2gh.
  • #1

Homework Statement


A boy throws a rock at speed v_0 at angle α from a balcony of height h. When the ball hits the ground, its velocity makes an angle θ with the ground. What is the minimum value of θ in terms of h and v_0


Homework Equations



basic kinematics equations.


The Attempt at a Solution



I started out with the following:

[tex] v_x = v_0 \cos \alpha , v_y = \sqrt{v_0^2\sin^2 \alpha+2gh} [/tex]

Then:

[tex] f( \theta ) = \tan^{-1} (\frac{\sqrt{v_0^2 \sin^2 \alpha+2gh}}{v_0\cos\alpha} [/tex]

To find the minimum, you have to differentiate, although whenever I do this i get a HUGE expression which doesn't make sense. Any help?Thanks :)
 

Attachments

  • Screen Shot 2013-09-14 at 11.18.41 AM.png
    Screen Shot 2013-09-14 at 11.18.41 AM.png
    4.4 KB · Views: 394
Last edited:
Physics news on Phys.org
  • #2
When ## \theta ## is max, ## \tan \theta ## is max as well.
 
  • #3
But we need to minimize theta?
Can you please give another hint?
 
  • #4
Ah, sorry. I really meant to say min, not max :)
 
  • #5
Gotcha.

So am I correct in saying that after taking the tangent of both sides, the RHS equals zero? in which case :

[tex] v_0^2 \sin^2 \alpha = 2gh [/tex] ??

I attempted it again, however i cannot get rid of alpha here.
 
  • #6
You need ## (\tan \theta)' = 0 ##.
 
  • #7
I still don't get it. Why is that so? Thanks.
 
  • #8
Let ## f(x) ## be a monotonic function. Let ## g(x) ## be any function. Consider ## F(x) = f(g(x)) ##. ## F'(x) = (f(g(x)))' = f'(g(x)) g'(x) = 0 ##. Because ## f(x) ## is monotonic, ## f'(x) \ne 0 ##, thus the previous equation implies ## g'(x) = 0 ##, which means that if ## F(x) ## has a stationary point, ## g(x) ## has a stationary point there as well.

## \tan ## is monotonic from zero to ## \pi/2 ##.
 
  • #9
So does this mean I have to differentiate [tex] \frac{\sqrt{v_0^2\sin^2\alpha + 2gh}}{v_0\cos\alpha}[/tex]?
Also, what do i differentiate with respect to? And how do i get rid of alpha? sorry I'm very stuck on this
 
  • #10
Since you are required to find that in terms of ##h## and ##v_0##, ##\alpha## has to disappear, which apparently means the minimum must be with respect to it.
 

1. What is the formula for finding the minimum θ for a rock thrown at speed v0?

The formula for finding the minimum θ is θ = arctan(v02/gR), where v0 is the initial speed of the rock, g is the acceleration due to gravity, and R is the radius of the circular path the rock must follow.

2. How does the initial speed of the rock affect the minimum θ?

The initial speed of the rock, v0, directly affects the minimum θ. As v0 increases, the minimum θ decreases. This means that the rock must be thrown at a higher initial speed in order to achieve a smaller minimum angle.

3. What is the significance of the minimum θ in this scenario?

The minimum θ represents the angle at which the rock must be thrown in order to achieve the farthest distance. Any angle above the minimum θ will result in a shorter distance, while any angle below the minimum θ will result in the rock falling short of the desired distance.

4. Can the minimum θ be negative or greater than 90 degrees?

No, the minimum θ cannot be negative or greater than 90 degrees. This is because the minimum θ is the angle at which the rock must be launched in order to achieve the farthest distance, and launching the rock at a negative angle or an angle greater than 90 degrees would not result in a valid trajectory.

5. Does the minimum θ change if the radius of the circular path is altered?

Yes, the minimum θ will change if the radius of the circular path is altered. As the radius increases, the minimum θ will also increase. This is because a larger radius requires a greater angle in order for the rock to travel the same distance.

Suggested for: Finding the Minimum θ for Rock Thrown at Speed v_0

Back
Top