Max height of a projectile equal to range?

In summary, the angle necessary to equal the range and the max height of a projectile is 63.43 degrees.
  • #1
puck
3
0
Hey, I'm having a problem determining the angle necessary for
the range of a projectile to equal the max height given the velocity
of the projectile. (the velocity is 53.1m/s)

I'd imagine it's necessary to set the equation for max height
equal to the range? Either way, I'm dumbfounded on how exactly
to obtain the angle of the projectile to have max height = range.

Any help would be greatly appreciated [I'm just a beginning
Physics nerd!]
-Steve
 
Last edited:
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  • #2
Is the answer 63.43 degrees?

I used these equations:

vox*t = voy*t - .5gt^2
vf = voy + at
with a = -g [due to gravity being negative]
vf = voy - gt
0 = voy - gt
since final velocity at peak height is zero
t = voy/g

Then subsitute:

2(vox*voy/g) = 2(voy^2/g) - g*(voy^2/g^2)
2vox*voy = 2voy^2 - voy^2
2vox*voy = voy^2
2vox = voy
2 = voy/vox

and since voy = v*sin theta
vox = v*cos theta

then 2 = v*sin theta/v*cos theta
v's cancel

2 = sin theta/ cos theta
and since tan = sin/cos
2 = tan theta

arc tan 2 = theta
which is 63.43 degrees

Is that right?

Anyone?
 
  • #3
I got a different answer. You may have accidently used the same time for both max height and range!

vy= v0 sin(θ)- gt= 0 at max height to
t= v0 sin(θ)/g .

Also y= v0 sin(θ)t- g/2 t2 is the height so the maximum height is (v02/g) sin2(θ)- (v02/2g)sin2(θ)= (v02/2g)sin2(θ).

x= v0 cos(θ)t so the range is v0 cos(θ)(2v0 sin(&theta)/g) (Notice the "2". Since we are neglecting air resistance, the motion is symmetrical. The projectile hits the ground in TWICE the time it takes to get to its maximum height.)
Range= (2 v02/g)sin(&theta)cos(&theta).

Setting range equal to maximum height,

(2 v02/g)sin(&theta)cos(&theta)= (v02/2g)sin2(θ).


v0 cancels (initial speed is not relevant!). We can also cancel the "g" terms. θ= 0 is an obvious solution: if we fire at angle 0 both maximum height and range are 0! Assuming sin(θ) is not 0, we can divide both sides of the equation by that and have

1/2 sin(θ)= 2 cos(θ) or tan(θ)= 4 so θ= 76 degrees.

(If you neglect the fact that the projectile hits the ground again in twice the time to max height, you get tan(θ)= 2 and that gives your answer.)
 
  • #4
Ahh, that makes perfect sense now! I thought I had the angle wrong, as it was a rather small angle [and the range and max height were not the same].

Thanks!
-Steve
 

1. What is the "max height" of a projectile?

The "max height" of a projectile refers to the highest point that the object reaches during its flight.

2. How is the "max height" of a projectile calculated?

The "max height" of a projectile can be calculated using the formula: max height = (initial velocity)^2 * sin^2(theta) / 2 * gravity, where theta is the launch angle and gravity is the acceleration due to gravity.

3. What is the relationship between the "max height" and the range of a projectile?

The "max height" and the range of a projectile are equal when the launch angle is 45 degrees. This is because the horizontal and vertical components of the projectile's motion are equal at this angle, resulting in an equal max height and range.

4. How does air resistance affect the "max height" of a projectile?

Air resistance can decrease the "max height" of a projectile by reducing its velocity and causing it to lose energy to air resistance. This is especially true for objects with a larger surface area, as they experience greater air resistance.

5. Can the "max height" of a projectile ever be greater than the range?

No, the "max height" of a projectile can never be greater than the range. This is because the projectile's horizontal velocity remains constant, and it will always cover the same horizontal distance regardless of its vertical motion.

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