Calculating velocity of a projectile accounting for drag

[batsali]

Hi,

I've been toying around with a cheap slingshot and some steel balls and cannot figure out how to calculate the velocity of the projectile at a given time/distance.

What I've done so far:

I derived a fairly accurate formula for the potential energy of the slingshot by measuring the pull force at several points and using my graphing calculator to integrate F(d) = 114.2d^0.678 from 0 to 0.5 meters find the total energy which is around 21 J.

I can easily find the initial velocity of the steel ball (assuming all other parts of the slingshot are massless), and calculate the force of air drag at that instant. However, I don't know calculus and cannot derive a formula that will give me the instantaneous acceleration or velocity with the ever decreasing drag. I've tried doing it without differential equations but the deceleration then turns out to be linear, which I don't think is correct.

I would really appreciate someone giving me a hand in this endeavor, because I've been reading calculus materials for the most part of today, but cannot seem to understand enough to enable me do solve my problem. Here is the data:

mass of ball - 0.0035 kg
sectional area - 7.088e-5 m2
KE - 21 J
air density - 1.125 kg/m3
Cd - 0.47

Also, could you tell me how to use differential equations with a TI-84 calculator

Thank you

 

[LightHero]

for your help!Calculating the velocity of a projectile at a given time or distance can be done using calculus. Specifically, you can use Newton's second law, which states that the acceleration of an object is equal to the sum of all forces acting upon it. This can be written as: F = maWhere F is the sum of all forces, m is the mass of the object, and a is its acceleration. In this case, the sum of forces includes the force of gravity (Fg), the force of air resistance (Fa), and the initial kinetic energy (KE). This equation can then be written as follows: Fg - Fa = maAssuming that the ball is released from rest, the initial kinetic energy can be written as: KE = (1/2)mv0^2 Where v0 is the initial velocity of the ball. The force of gravity can be written as: Fg = mg Where g is the acceleration due to gravity. The force of air resistance can be written as: Fa = (1/2)ρav^2Cd Where ρ is the air density, a is the sectional area of the ball, v is the instantaneous velocity of the ball, and Cd is the drag coefficient. Finally, you can use this equation to calculate the acceleration of the ball as a function of time, which can then be integrated to find the velocity of the ball at any given time. To do this on a TI-84 calculator, you first need to enter the equation into the "Y=" screen. You can then graph the equation and examine the graph to determine the velocity of the ball at a given time. Alternatively, you can also use the calculator's built-in numerical integration function to integrate the equation and find the velocity of the ball at a given time.

 

[astrokreng]

for reaching out and sharing your project with us. Calculating the velocity of a projectile accounting for drag can be a challenging task, but I am happy to assist you in any way I can.

To start, let's break down the problem into smaller parts. First, we need to understand the forces acting on the projectile. In this case, we have the initial force from the slingshot, the force of gravity, and the force of air resistance (drag). The initial force from the slingshot can be calculated using your derived formula for potential energy. The force of gravity can be calculated using the mass of the projectile and the acceleration due to gravity (9.8 m/s^2).

The force of air resistance, also known as drag, is a bit more complicated. It is dependent on the velocity of the projectile, the air density, the cross-sectional area, and the drag coefficient. The drag coefficient (Cd) is a dimensionless number that represents the shape and size of the object and its interaction with the air. In your case, you have provided a value of 0.47, which is the drag coefficient for a spherical object.

To calculate the force of air resistance, we can use the following formula:

Fdrag = 0.5 * Cd * air density * velocity^2 * cross-sectional area

Now, let's move on to the differential equations. Differential equations are used to describe the relationship between a function and its derivatives. In this case, we can use the equation of motion for a projectile with air resistance to describe the velocity of the projectile over time. This equation is:

dv/dt = (Fslingshot - Fdrag - Fgravity) / m

where dv/dt is the instantaneous acceleration, Fslingshot is the force from the slingshot, Fdrag is the force of air resistance, Fgravity is the force of gravity, and m is the mass of the projectile.

To use this equation with your TI-84 calculator, you will need to use a numerical method called Euler's method. This method uses small time intervals to approximate the solution to a differential equation. I would recommend doing some research on Euler's method and how to use it with your calculator, as it may be too complex to explain in this response.

In summary, to calculate the velocity of your projectile accounting for drag, you will need to use the formula for the force of air resistance, as well as the equation of