Projectile Motion and Air resistance

[usahockey7]

I'm taking grade 12 physics and I've been given a lab that involves a slingshot. A baseball will be shot using a slingshot and is to go 40m. I have to perform calculations to find out the initial velocity it needs to travel a distance of 40m, however, I have to take air resistance into account and I have no idea on how to do this, I tried writing my own program using java, but tragically, that didn't go over too well so I need a new approach. I have already found the spring constant as well as the efficiency of the spring, I just don't know how to take air resistance into account here.

 

[Fewmet]

I'm taking grade 12 physics and I've been given a lab that involves a slingshot. A baseball will be shot using a slingshot and is to go 40m. I have to perform calculations to find out the initial velocity it needs to travel a distance of 40m, however, I have to take air resistance into account and I have no idea on how to do this, I tried writing my own program using java, but tragically, that didn't go over too well so I need a new approach. I have already found the spring constant as well as the efficiency of the spring, I just don't know how to take air resistance into account here.

I recommend focusing on the work done by friction and its effect of the energy of the ball.

Do you see how to do that?

 

[usahockey7]

I recommend focusing on the work done by friction and its effect of the energy of the ball.

Do you see how to do that?

The work done by the air on the ball, which will be negative work because it's in the opposite direction of the ball's motion? My teacher told us to calculate the air resistance and use an fbd or something and calculate the new acceleration and velocity for the ball during each second of flight? I'm still lossed on where to start. I only have the k value and the efficiency of the spring so far, maybe if i started with a random velocity and then worked from there, but that would take forever, is there a formula that relates air resistance and distance to initial velocity? If I worked backwards it would be so much easier

 

[Fewmet]

The work done by the air on the ball, which will be negative work because it's in the opposite direction of the ball's motion?

That is correct. If you had a value for the force of friction, you could simply multiply it by the distance traveled and that that work decreases the kinetic energy by the same amount.

My teacher told us to calculate the air resistance and use an fbd or something and calculate the new acceleration and velocity for the ball during each second of flight?

If those are the instructions, you should ignore the suggestion about energy.

If you have a value for the force of friction, you need just apply the second law ($$\Sigma$$F=ma) to find the acceleration.

I'm still lossed on where to start. I only have the k value and the efficiency of the spring so far, maybe if i started with a random velocity and then worked from there, but that would take forever, is there a formula that relates air resistance and distance to initial velocity? If I worked backwards it would be so much easier

Can you say what the k value means? Is it a spring constant?

I'm not seeing what to do to get the force of friction.

 

[usahockey7]

That is correct. If you had a value for the force of friction, you could simply multiply it by the distance traveled and that that work decreases the kinetic energy by the same amount.



If those are the instructions, you should ignore the suggestion about energy.

If you have a value for the force of friction, you need just apply the second law ($$\Sigma$$F=ma) to find the acceleration.




Can you say what the k value means? Is it a spring constant?

I'm not seeing what to do to get the force of friction.

Yeah the k value in this case is the spring constant of the slingshot

 

[usahockey7]

If we ignore air resistance. I would find the initial speed of the ball using one of the kinematic equations for the ball to travel 40m, then use that initial speed to find the object's kinetic energy at the moment it leaves the slingshot, then using the efficiency of the spring or slingshot in this case, find the elastic potential by multiplying the efficiency by the kinetic energy and then solve for x which is the distance the sling shot has to be pulled back to achieve the required initial velocity for the ball to travel 40m, however, this lab requires that air resistance is taken into consideration and my class has never been introduced to air resistance before so I have no idea on how to take this into consideration

 

[jhae2.718]

This may be getting a little advanced for a 12th grade course, but if you're interested:

For the air resistance, you're probably going to have to numerically solve an ordinary differential equation.

The magnitude of air drag is $$F_D = C_D \frac{1}{2}\rho V^2 S$$. Here, $C_D$ is the coefficient of drag, $\rho$ is air density, V is the velocity of the object, and S is the cross sectional area. To simplify things, we'll write the drag force as $F_D = bV^2$, where $b= 0.5\rho C_D S$.

This force will act opposite the direction of motion in both the x and y components. If you write the velocity as a vector, $\mathbf{v} = v_x \hat{\imath} + v_y \hat{\jmath}$, you can write the drag force components as [itex]F_{Dx} = -bVV_x[/tex] and $F_{Dy} = -bVV_y$.

Next, get your equations of motion from the free body diagram. You will have two differential equations which you can solve to find the x and y positions as a function of time.
You'll want to solve these numerically. You can solve this with a programming language or with a spreadsheet program. (The RK4 solver should be sufficient, and you should be able to find an open source implementation in Java.)

 

[Fewmet]

If we ignore air resistance. I would find the initial speed of the ball using one of the kinematic equations for the ball to travel 40m, then use that initial speed to find the object's kinetic energy at the moment it leaves the slingshot, then using the efficiency of the spring or slingshot in this case, find the elastic potential by multiplying the efficiency by the kinetic energy and then solve for x which is the distance the sling shot has to be pulled back to achieve the required initial velocity for the ball to travel 40m, however, this lab requires that air resistance is taken into consideration and my class has never been introduced to air resistance before so I have no idea on how to take this into consideration

usahockey7: If what jhae2.718 posted is accessible, it's the the way to go.

If not, do you think it would be within the confines of the assignment to do a trial run, see how far short of 40 m the ball falls and use that refine the prediction?

Listen: when this is all done and you get of sense of what you teacher thought was a reasonable approach to take, can you post that here?

 

[usahockey7]

usahockey7: If what jhae2.718 posted is accessible, it's the the way to go.

If not, do you think it would be within the confines of the assignment to do a trial run, see how far short of 40 m the ball falls and use that refine the prediction?

Listen: when this is all done and you get of sense of what you teacher thought was a reasonable approach to take, can you post that here?

I found out today and the above suggestions about programming yourself is the way to go, I'm in a group but the other members aren't doing anything so it looks like I'm on my own, and every group was assigned with a "programmer" however, it's evident that my group obviously has no programming experience lol, I guess if there's no other suggestions, I'm just going to use free body diagrams to solve for the acceleration and then solve for the velocity of the ball during each second or half of a second during its flight.

 

[jhae2.718]

What is the highest level of math you have taken?

 

[usahockey7]

What is the highest level of math you have taken?

I'm currently in grade 12, so I've only taken pre calc and advanced functions, though I'm going to Waterloo for mech engineering in the fall.

 

[jhae2.718]

Did your teacher explain any of the programming needed to solve the diff eq.? If you've only had pre-calc, asking you to numerically solve a nonlinear ordinary differential equation is a bit much.

On that note, did your teacher want you to use the drag equation, or did he give a simpler equation such as F = bv? (Some books will model drag as proportional to velocity to simplify things, even though it is proportional to the square of velocity.)

 

[usahockey7]

Did your teacher explain any of the programming needed to solve the diff eq.? If you've only had pre-calc, asking you to numerically solve a nonlinear ordinary differential equation is a bit much.

On that note, did your teacher want you to use the drag equation, or did he give a simpler equation such as F = bv? (Some books will model drag as proportional to velocity to simplify things, even though it is proportional to the square of velocity.)

I think he wants us to use the drag equation, but I've heard it's not a linear function? And it's difficult to work with, without using a program? The easiest way to solve it I guess is to use a program but I have no knowledge of programming, I've never taken a programming course throughout high school.

 

[jhae2.718]

And it's difficult to work with, without using a program?

Yes. If there is an easy way to solve the DEs analytically, I'm not aware of it.

The easiest way to solve it I guess is to use a program but I have no knowledge of programming, I've never taken a programming course throughout high school.

Right. Are there any languages you are familiar with? I prefer MATLAB for doing numerical analysis, but I doubt you'll have access to it. There is a free clone called Octave that is similar to MATLAB. Other good candidates are NumPy, a numerical version of Python (I'm only tangentially familiar with it) and Sage.

I think he wants us to use the drag equation, but I've heard it's not a linear function?

No, the drag equation is a quadratic function. When you solve the equations of motion you get these second order nonlinear ordinary differential equations (bear with me for a moment):
$$\begin{align*} \ddot{x}&= -\frac{b}{m}vv_x \\ \ddot{y}& = -g - \frac{b}{m}vv_y \end{align*}$$
The double dots on x and y mean the second derivative of x and y, which is acceleration. A derivative, if you are unfamiliar with them (I'm not sure what's covered in your pre-cal class, some will introduce derivatives at the end), gives the rate of change of a function. It is essentially the "slope" of a function.

Now, the problem here is that to solve these equations numerically we need them in the form $y' = f(t,y)$. This means we will need to solve four equations simultaneously. First, we find velocity:
$$\begin{align*} \dot{v} _x&= -\frac{b}{m}vv_x \\ \dot{v} _y&= -g - \frac{b}{m}vv_y \end{align*}$$
Then we use the velocity to find the position:
$$\begin{align*} \dot{x} &= v_x(t) \\ \dot{y} &= v_y(t) \\ \end{align*}$$

Then we use a solver to find the solutions to these equations. We'll need initial conditions, which we'll get from your work on the earlier part of the problem.

 

[usahockey7]

Yes. If there is an easy way to solve the DEs analytically, I'm not aware of it.


Right. Are there any languages you are familiar with? I prefer MATLAB for doing numerical analysis, but I doubt you'll have access to it. There is a free clone called Octave that is similar to MATLAB. Other good candidates are NumPy, a numerical version of Python (I'm only tangentially familiar with it) and Sage.


No, the drag equation is a quadratic function. When you solve the equations of motion you get these second order nonlinear ordinary differential equations (bear with me for a moment):
$$\begin{align*} \ddot{x}&= -\frac{b}{m}vv_x \\ \ddot{y}& = -g - \frac{b}{m}vv_y \end{align*}$$
The double dots on x and y mean the second derivative of x and y, which is acceleration. A derivative, if you are unfamiliar with them (I'm not sure what's covered in your pre-cal class, some will introduce derivatives at the end), gives the rate of change of a function. It is essentially the "slope" of a function.

Now, the problem here is that to solve these equations numerically we need them in the form $y' = f(t,y)$. This means we will need to solve four equations simultaneously. First, we find velocity:
$$\begin{align*} \dot{v} _x&= -\frac{b}{m}vv_x \\ \dot{v} _y&= -g - \frac{b}{m}vv_y \end{align*}$$
Then we use the velocity to find the position:
$$\begin{align*} \dot{x} &= v_x(t) \\ \dot{y} &= v_y(t) \\ \end{align*}$$

Then we use a solver to find the solutions to these equations. We'll need initial conditions, which we'll get from your work on the earlier part of the problem.

Perfect, that made sense, so how would I go about solving for the initial velocity needed for the ball to travel 40m? This is where I'm stuck. Obviously without air resistance I could use the range equation, is there a way to substitute the equation for air resistance into the range equation to find the initial velocity required to shoot the ball 40m with the slingshot, because from there i can just find the kinetic energy, mulitply it by the efficiency of the spring and then find x using hooke's spring law.

 

[jhae2.718]

The difficulty here is that, because air resistance is dependent on velocity, there's no easy analytic solution. I can't think of a way to analytically determine the initial conditions. Perhaps another homework helper knows how, but I think you're going to have to determine the initial velocity by playing with the initial conditions. (It's not as hard as it sounds; a good initial guess is to solve the problem ignoring air resistance.)