Football Trajectory: Radius & Speed Rate of Change

[bkw2694]

Homework Statement

A football player releases a ball at 35° with initial velocity of 80 ft/sec. Determine the radius of curvature of the trajectory at times t = 1 sec and t = 2 sec, where t = 0 is the time of release from the quarterback's hand. For each case, compute the time rate of change of the speed.

Book answers: at t = 1: p = 142.2 ft, a_t = -6.58 ft/sec^2
at t = 2: p = 149.7 ft, a_t = 8.75 ft/sec^2

Homework Equations

a_n = v^2/ρ
a_t = dv/dt = acceleration in tangential direction
v = ρ(dΘ/dt)
a = dv/dt

The Attempt at a Solution

I'm really confused about how to tackle a problem like this that involves specific times. In class and in all the homework up to this point we haven't had an example of how to approach these types of problems, but I'm guessing there's a derivative or integral that I'm not seeing or understanding that's killing me, but the closest thing I could come up with is:

magnitude of acceleration = g = 32.2 ft/sec^2
v = ρ(dΘ/dt), which I split into: v dt = ρ dΘ.
I then made an integral: ∫(from 0 to t) 80 dt = ρ ∫(from 0 to 35*π/180) dΘ.
This gave me 80t = (.6109)ρ, when t = 1 gives me ρ = 131 ft.
I then tried solving for a_n with v^2/ρ, which gave me a_n = 48.87, which can't be possible as it makes magnitude a = g false...

Thanks in advance.

 

[SteamKing]

Homework Statement

A football player releases a ball at 35° with initial velocity of 80 ft/sec. Determine the radius of curvature of the trajectory at times t = 1 sec and t = 2 sec, where t = 0 is the time of release from the quarterback's hand. For each case, compute the time rate of change of the speed.

Book answers: at t = 1: p = 142.2 ft, a_t = -6.58 ft/sec^2
at t = 2: p = 149.7 ft, a_t = 8.75 ft/sec^2

Homework Equations

a_n = v^2/ρ
a_t = dv/dt = acceleration in tangential direction
v = ρ(dΘ/dt)
a = dv/dt

The Attempt at a Solution

I'm really confused about how to tackle a problem like this that involves specific times. In class and in all the homework up to this point we haven't had an example of how to approach these types of problems, but I'm guessing there's a derivative or integral that I'm not seeing or understanding that's killing me, but the closest thing I could come up with is:

magnitude of acceleration = g = 32.2 ft/sec^2
v = ρ(dΘ/dt), which I split into: v dt = ρ dΘ.
I then made an integral: ∫(from 0 to t) 80 dt = ρ ∫(from 0 to 35*π/180) dΘ.
This gave me 80t = (.6109)ρ, when t = 1 gives me ρ = 131 ft.
I then tried solving for a_n with v^2/ρ, which gave me a_n = 48.87, which can't be possible as it makes magnitude a = g false...

Thanks in advance.

You are assuming that the football follows a circular arc after it leaves the player's hand. This may not be a valid assumption.

BTW, there is a mathematical definition of radius of curvature, which you should know:

http://mathworld.wolfram.com/RadiusofCurvature.html

I would evaluate the football's trajectory just like any other projectile motion problem, and apply the definition of radius of curvature as given in the linked article.

 

[TSny]

I would determine a_n first, and then use a_n = v²/ρ to get the radius of curvature.

 

[bkw2694]

You are assuming that the football follows a circular arc after it leaves the player's hand. This may not be a valid assumption.

BTW, there is a mathematical definition of radius of curvature, which you should know:

http://mathworld.wolfram.com/RadiusofCurvature.html

I would evaluate the football's trajectory just like any other projectile motion problem, and apply the definition of radius of curvature as given in the linked article.

So if it isn't following a circular arc, this means that ρ is changing, correct? Or is it that Θ is changing and causing ρ to change?

Also, I see the radius of curvature equation is: [1+(dy/dx)^2]^(3/2) / [(d^2y/dx^2)]. But I don't understand how to make that apply to this particular question since I'm not given any equation for y(x) or x(y).

 

[SteamKing]

So if it isn't following a circular arc, this means that ρ is changing, correct? Or is it that Θ is changing and causing ρ to change?

Like most projectiles thrown in a gravity field (ignoring air resistance), the ball's trajectory is going to be a parabola, rather than a circle. This is why I suggested that the path of the ball be analyzed like projectiles in general, without making unnecessary assumptions about the trajectory.

In short, the R.O.C. of the ball's path will be changing w.r.t. time and will not be a constant like it would if its path were in fact circular.

Also, I see the radius of curvature equation is: [1+(dy/dx)^2]^(3/2) / [(d^2y/dx^2)]. But I don't understand how to make that apply to this particular question since I'm not given any equation for y(x) or x(y).

Pay attention to the formula for the radius of curvature where x and y are functions of time, IOW, x = x(t) and y = y(t).

By analyzing the trajectory of the football, you should be able to determine the coordinates of the ball's trajectory at the times requested, along with any time derivatives which are required to calculate the R.O.C.

 

[bkw2694]

Like most projectiles thrown in a gravity field (ignoring air resistance), the ball's trajectory is going to be a parabola, rather than a circle. This is why I suggested that the path of the ball be analyzed like projectiles in general, without making unnecessary assumptions about the trajectory.

In short, the R.O.C. of the ball's path will be changing w.r.t. time and will not be a constant like it would if its path were in fact circular.
Pay attention to the formula for the radius of curvature where x and y are functions of time, IOW, x = x(t) and y = y(t).

By analyzing the trajectory of the football, you should be able to determine the coordinates of the ball's trajectory at the times requested, along with any time derivatives which are required to calculate the R.O.C.

Yeah, sorry, I just now realized what you meant and used the normal projectile equations to find ρ. By using the correct formula, I ended up getting ρ at t=1 is 142.19 ft.

So now the only problem I'm having is I can't seem to work out the correct a_t. Am I wrong in assuming that the magnitude of acceleration = g? Or does that change as well? Because I tried solving for a_n with 80^2/142.19 = 45.01, then substituting that into a = sqrt(an^2 + at^2) and it's giving me a way off answer.

 

[TSny]

What angle does the velocity vector make to the horizontal at t = 1 s?
What angle does the acceleration vector make to the normal direction at t = 1 s?

Am I wrong in assuming that the magnitude of acceleration = g?

Yes, the magnitude of the acceleration vector must be g.

 

[SteamKing]

Yeah, sorry, I just now realized what you meant and used the normal projectile equations to find ρ. By using the correct formula, I ended up getting ρ at t=1 is 142.19 ft.

So now the only problem I'm having is I can't seem to work out the correct a_t. Am I wrong in assuming that the magnitude of acceleration = g? Or does that change as well? Because I tried solving for a_n with 80^2/142.19 = 45.01, then substituting that into a = sqrt(an^2 + at^2) and it's giving me a way off answer.

Now that you have calculated the R.O.C. of the ball's path at a particular time, you can assume for a brief instant that its path approximates that of a circle with a radius = R.O.C. In that case, knowing the tangential velocity and the R.O.C., you should be able to use the circular motion equations and calculate the tangential acceleration of the ball.

 

[bkw2694]

What angle does the velocity vector make to the horizontal at t = 1 s?
What angle does the acceleration vector make to the normal direction at t = 1 s?

Yes, the magnitude of the acceleration vector must be g.

Okay, so I can see that the angle will be getting smaller, but I'm not sure what equation to use to figure this one out. Is this just a projectile equation I'm overlooking or is there calculus involved here?

Since I knew the answer from the book I worked backwards and found an, then solved to see what the velocity should be, which is 66.96 ft/sec.

 

[haruspex]

Okay, so I can see that the angle will be getting smaller, but I'm not sure what equation to use to figure this one out. Is this just a projectile equation I'm overlooking or is there calculus involved here?

Since I knew the answer from the book I worked backwards and found an, then solved to see what the velocity should be, which is 66.96 ft/sec.

In case it is not clear from the other posts, you are asked for the rate of change of speed, whereas g is the magnitude of the rate of change of velocity. For the rate of change of speed you need to restrict to the tangential acceleration. How might you derive that from g?

 

[bkw2694]

In case it is not clear from the other posts, you are asked for the rate of change of speed, whereas g is the magnitude of the rate of change of velocity. For the rate of change of speed you need to restrict to the tangential acceleration. How might you derive that from g?

Okay, so I used trigonometry to find a_t = gcos(Θ). And I suppose I can use the same thing to find an = -gsin(Θ). I tried plugging those into a = sqrt(an^2 + at^2), but I think that causes the Θ to disappear due to sin^2 + cos^2 = 1.

 

[bkw2694]

^whoops, I mixed those two up, at = -gsin(Θ); an = gcos(Θ)

 

[TSny]

It's basically just a projectile motion problem with the additional formula a_n = v²/ρ. You know the magnitude of the acceleration. Therefore, you can find the normal component if you know the angle between the acceleration vector and the normal direction. How is the normal direction related to the velocity direction? How can you get the direction of the velocity vector?

 

[Chestermiller]

So if it isn't following a circular arc, this means that ρ is changing, correct? Or is it that Θ is changing and causing ρ to change?

Also, I see the radius of curvature equation is: [1+(dy/dx)^2]^(3/2) / [(d^2y/dx^2)]. But I don't understand how to make that apply to this particular question since I'm not given any equation for y(x) or x(y).

If x = x(t) and y = y(t), show using the above equation for the radius of curvature that:

$$R=\frac{\left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right]^{\frac{3}{2}}}{\left[\frac{dx}{dt}\frac{d^2y}{dt^2}-\frac{dy}{dt}\frac{d^2x}{dt^2}\right]}$$

Chet

 

[bkw2694]

I must be solving for v (when t = 4) incorrectly. a_n = v^2/p, but when I try doing v = 80cos(theta), then plug gcos(theta) = (80cos(theta))^2/142.2 I end up with theta = 44.3 degrees, but I know the correct answer should be more like 11 degrees.

 

[TSny]

I must be solving for v (when t = 4) incorrectly.

Did you mean t = 1?

a_n = v^2/p, but when I try doing v = 80cos(theta), then plug gcos(theta) = (80cos(theta))^2/142.2 ...

I don't follow what you are doing here. In the equation a_n = v²/ρ, v is the speed of the particle: v = (v_x²+ v_y²)^1/2.
So you can get v at t = 1 s if you first determine v_x and v_y at t = 1 s using standard projectile motion equations.

 

[bkw2694]

Did you mean t = 1?

I don't follow what you are doing here. In the equation a_n = v²/ρ, v is the speed of the particle: v = (v_x²+ v_y²)^1/2.
So you can get v at t = 1 s if you first determine v_x and v_y at t = 1 s using standard projectile motion equations.

Sorry, I one of my other HW problems had t = 4 so I screwed that up.

Okay, I figured out my problem. Aside from using the wrong time, I was also using the wrong v. I was using v = 80cosθ, but v was obviously 80 ft/sec and provided as the initial velocity. The incorrect v value with cosθ was giving me some crazy answers and really throwing me off.

Huge thanks to everyone in this thread for bearing with me on this one. Now that I see how easy this problem actually was I'm really sorry for taking up everyone's time, but I really appreciated it!