Hello

I am doing some work on projectiles and would like some help. I am
embarassed to say that I cannot rearrange some equations. I always end
up with the form T^2 + TX = Y and cannot solve to get T by itself.
Somthing of the form T = .... The equations I am trying to rearrange
are the projectile equations. In the book Physics for game developers
there is a list of equations on page 104 and 105 which describe the
speed, total time etc for projectiles being fired from a platform to a
platform above and an platform below. These equation assume you know
the initial velocity and the angle of firing.
What I plan to do is know the distance to these two platfroms (one
above and the other below the firing platform) and know the angle of
firing. What I am trying to do is rearrange the equations so that I
fing the intial horizontal speed and the total timme of flight. This
is whre I run into trouble. Does anyone know the equations that can
help me? I have got them where you know speed and angle. I wnat them
in the form that you know the distacne between platfrom and and angle.

Thanks

Richard

Projectile
http://scienceworld.wolfram.com/physics/Projectile.html

Trajectory
http://scienceworld.wolfram.com/physics/Trajectory.html

Rich wrote in message
om...
Hello

I am doing some work on projectiles and would like some help. I am
embarassed to say that I cannot rearrange some equations. I always end
up with the form T^2 + TX = Y and cannot solve to get T by itself.
Somthing of the form T = .... The equations I am trying to rearrange
are the projectile equations. In the book Physics for game developers
there is a list of equations on page 104 and 105 which describe the
speed, total time etc for projectiles being fired from a platform to a
platform above and an platform below. These equation assume you know
the initial velocity and the angle of firing.
What I plan to do is know the distance to these two platfroms (one
above and the other below the firing platform) and know the angle of
firing. What I am trying to do is rearrange the equations so that I
fing the intial horizontal speed and the total timme of flight. This
is whre I run into trouble. Does anyone know the equations that can
help me? I have got them where you know speed and angle. I wnat them
in the form that you know the distacne between platfrom and and angle.

Thanks

Richard


The given equation: T^2 + TX = Y (dimensionally incorrect)
is of the form a*T^2 + b*T + c = 0
wherein a = 1, b = X, and c = -Y
The general solution is givem by

T = [-b + sqrt( b^2 - 4ac )] / 2a
or T = [-b - sqrt( b^2 - 4ac )] / 2a

For positive T we get

2T = -X + sqrt( X^2 + 4Y )

Diffentiating the equation given by Richard WRT T yields

2T + [ X + TV_x ] = V_y or T = [ V_y - X ] / [V_x + 2]

where V_x = dX/dT and V_y = dY/dT are X and Y velocities.
This equation is consistant with V_y = 0 at T = X = 0.

Differentiating WRT T again yeilds

2 + [ 2V_x + TA_x ] = A_y = g

Where A_x and A_y are X and Y accelerations. A_y = g is the
acceleration of gravity. For a projectile in vacuum A_x = 0 for
all T. This yields

1 + V_x = g / 2

Which is a nonsensical constraint (also dimensionally incorrect).
[Old Man]

(Rich) wrote in message . com...
Hello

I am doing some work on projectiles and would like some help. I am
embarassed to say that I cannot rearrange some equations. I always end
up with the form T^2 + TX = Y and cannot solve to get T by itself.
Somthing of the form T = ....

I'll bet you're one of those people who said, at the
end of high-school algebra "I'll bet nobody in the
history of the universe has ever actually needed to
know the quadratic formula. Why did I learn that?"

You have a quadratic equation here for T. Guess what
formula you need for the solution to T?

Here's the answer in your terms. If you have
T^2 + TX = Y, rearrange to standard form
T^2 + TX - Y = 0. There are two solutions:

T = -X +- sqrt(X^2+4Y)

Usually one of these two solutions can be dismissed
as unphysical (for instance referring to a negative
time).

- Randy

"Rich" wrote in message
om...
Thanks for the replies.

However I have found out today that my orig equation T^2 ..... is
actually wrong.
Let me explain what I am after.

I am firing a projectile
I know the position it starts and its end position. The end position
will be either above or below the start position. This amount is
known. that is I know (Xinit, Yinit) amd (Xfinal, Yfinal)
I also know the angle that the projectile will be fired at.

What I would like to know are formulae that allow me to work out the
init firing velocity and the total time of flight.

I have looked at the links given in previous post and they all have
equations going the other way. That is you know the angle and the
firign velocity and they work out the final position and tiem of
flight.

Does anyone know these equations???

Let d be the horizontal distance between starting and ending
position, h the vertical distance, and q the firing angle. Then
you should get expressions that look something like:

V = sqrt(g)*d/sqrt(2*cos(q)*(d*sin(q) - h*cos(q)))

t = sqrt(2*(d*sin(q) - h*cos(q))/g*cos(q))

Beware of signs for the variables.

Rich wrote in message
om...
Thanks for the replies.

However I have found out today that my orig equation T^2 ..... is
actually wrong.
Let me explain what I am after.

I am firing a projectile
I know the position it starts and its end position. The end position
will be either above or below the start position. This amount is
known. that is I know (Xinit, Yinit) amd (Xfinal, Yfinal)
I also know the angle that the projectile will be fired at.

What I would like to know are formulae that allow me to work out the
init firing velocity and the total time of flight.

I have looked at the links given in previous post and they all have
equations going the other way. That is you know the angle and the
firign velocity and they work out the final position and tiem of
flight.

Does anyone know these equations???

Thanks
Ricahrd

The physics are very simple: Two points, (t0, x0, y0) which is
the initial position and initial time, and, (t1, x1, y1) which is the final
position and final time. Let

X = x1 - x0
Y = y1 - y0
T = t1 - t0

The velocity in the x-direction is constant

Vx =Vx0

Therefore,

X = Vx0*T

The firing angle, A0, is given by

sin(A0) = Vy0 / V0
or Vy0 = V0*sin(A0)

cos(A0) = Vx0 / V0 = X / V0*T
or T = X / V0*cos(A0)

The acceleration of gravity, g, acts in the y-direction:

Y = Vy0*T - (g / 2)*T^2

Substituting for Vy0 and T from above yields

Y = X*tan(A0) - (g / 2)*T^2

Rearanging terms gives the desired equations for V0 and T:

T^2 = [ 2 / g ] [ X*tan(A0) - Y ]

V0 = X / T*cos(A0)

[Old Man]