Projectile motion and radius of curvature.

In summary: This was a tough one and I really appreciate all the help.In summary, the initial velocity is 16.57m/s.
  • #1
faust9
692
2
Ok, I'm vexed... I was given the following problem:

From measurements of a photograph it has been found that the stream of water leaving a nozzle at A had a radius of curvature of 35m.

(a) determine the initial velocity
(b) determine the radius of curvature at hmax

The water is shown leaving the nozzle along a 3-4-5 triangle where the horizontal component is 4 and the verticle component is 3 (or 36.1'ish degrees).

So, my question is how do I find velocity given an angle and a curvature? I know radius of curvature ([itex]\rho[/itex]) is:

[tex] \rho=\frac{[1+(y^{\prime})^2]^{2/3}}{y^{\prime \prime}}[/tex]

I have no idea where to go from here. My professor told us the answer was 16.57m/s but I haven't the foggiest clue on how to get there. Any help would be greatly appreciated.

Thanks.
 
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  • #2
Is a radius of curvature even defined? The water projects in a parabolic arc, not a circular arc. The radius of curvature changes constantly from the very beginning.
 
  • #3
JohnDubYa said:
Is a radius of curvature even defined? The water projects in a parabolic arc, not a circular arc. The radius of curvature changes constantly from the very beginning.

What do you mean defined? The initial curvature [itex]\rho=35m[/itex] was given in the problem. There is no functional definition of curvature but I dare say that ties in with the projectile motion equations in some manner. I just don't know how to do it. My text doesn't cover anything like this either. All of the projectile motion questions in the text are presented with straightforward initial conditions.
 
  • #4
JohnDubYa said:
Is a radius of curvature even defined? The water projects in a parabolic arc, not a circular arc. The radius of curvature changes constantly from the very beginning.

Yes, the radius of curvature is defined at each point, given by the formula faust9 gave. The radius of curvature is given at the nozzle and the question asks for the radius of curvature at the vertex of the parabola.

JohnDubYa's point that this is a parabola is important: set up a coordinate system so that the nozzle is at (0,0) and and the y-axis is parallel to the axis of the parabola. You can write the parabola as y= ax2+ bx. y'= 2ax+ b which, at x= 0, is y'= b. You can find b from the "3- 4- 5" information. y"= 2a and you find a from the curvature at x= 0.

Once you know the equation of the parabola you can use the fact y= -(g/2)t2+ vyt and x= vxt to find vx and vy[\sub] (the components of velocity at t= 0) to find the initial velocity.
 
  • #5
A million thanks.
 

1. What is projectile motion?

Projectile motion is the motion of an object through the air or space under the influence of gravity. It follows a curved path due to the gravitational force acting on it.

2. How is the radius of curvature related to projectile motion?

The radius of curvature is the radius of the circle that best approximates the curved path of a projectile. It is directly related to the velocity, acceleration, and angle of launch of the projectile.

3. Is the radius of curvature constant for all points along the trajectory of a projectile?

No, the radius of curvature changes at each point along the trajectory of a projectile. It is smallest at the highest point of the trajectory and increases as the projectile moves towards the ground.

4. How can the radius of curvature be calculated for a projectile?

The radius of curvature can be calculated using the formula R = v^2/g, where v is the velocity of the projectile and g is the acceleration due to gravity. This formula assumes that the angle of launch is 45 degrees.

5. How does air resistance affect the radius of curvature in projectile motion?

Air resistance can affect the radius of curvature in projectile motion by causing the projectile to slow down and deviate from its expected path, resulting in a larger radius of curvature. However, for most projectiles at low speeds, the effects of air resistance are negligible on the radius of curvature.

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