Re: How can you calculate the distance to what a projectile can go?

Date: Fri Mar 9 10:42:11 2001
Posted By: Samuel Silverstein, faculty, physics, Stockholm University
Area of science: Physics
ID: 982795061.Ph

Message:

Christian,

To answer your question, you need one more piece of information. Not only do you need to know the "spring constant" of your elastic, you also have to know how far back you pulled the elastic before letting go. With that information, we can proceed...

The main formula for the trajectory is quite simple:

Distance = Vx t

where Vx is the horizontal (x) velocity of the projectile, and t is the time that the projectile is in the air. Below, we will see how to calculate each of these terms.

Calculating Vx

Let's start with the elastic of your slingshot, and assume that it works in the same way as a spring. Then if you pull the elastic back a distance x, the magnitude of the force is:

F = kx

where k is the spring constant, measured in Newtons per meter. Ignoring for the moment the force of gravity (presumably, your slingshot provides far more force than gravity...), the kinetic energy of your projectile after it leaves the slingshot is:

    
E = 1/2 k x^2 = 1/2 mV^2

Solving for V, we get a velocity

V = x sqrt(k/m)

If the angle theta of the slingshot is measured from zero horizontally:

^ y
|
|  /
| / 
|/ theta
+-------------->x

then the horizontal velocity Vx is:

Vx = V cos(theta)

Calculating t

To calculate the time t that the projectile is in the air, you need to know its initial vertical velocity Vy. This is simply V times sin(theta). The expression for the vertical height of the projectile is:

y = yo + Vy t - 1/2 g t^2

where yo is the initial height (are you standing when you shoot the slingshot?), Vy is the inital y velocity, and g is the gravitational acceleration (9.8 m/s^2).

To find when the projectile hits the ground, solve thi expression for y = 0.

   0 = yo + Vy t - 1/2 g t^2

Using the quadratic method:

       -Vy +/- sqrt[Vy^2 + 4(1/2 g yo)]
  t =  --------------------------------
                -2(g/2)

       Vy      sqrt[Vy^2 + 2gyo]
    =  -- +/- ------------------ 
        g            g

Put in the values for yo, Vy, and g, and solve the above equation for both possible values of t. The value that is non-negative is the one you should combine with Vx from the first part to calculate the trajectory.

I hope this helps.

Regards,

Sam Silverstein

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