I am solving for a particular situation. A two vehicles approach an intersection and are on a collision course. To avoid the collision, one vehicle needs to apply the brakes such that it slows to allow the other vehicle to cross. There are two variables that control whether the application of brakes will be sufficient to avoid the collision: the time at which the brakes are applied, and the deceleration provided by the brakes.
Desmos tab demonstrating my approach x axis is time, y axis is distance. 0,0 is the accident location and time. y=ax is the formula for the distance over time of the vehicle assuming no braking is applied a = speed of vehicle distance/time or y/x y=-(a/(1-2d+d^2))(x-d)^2+ax The quadratic equation, shown in green, represents the deceleration of the vehicle due to brake application. d = time at which brakes are applied
Note that when x=d, the slope of the quadratic equation matches that of the linear equation, and it intersects there. This represents the vehicle applying brakes at that moment, then the distance/time, speed, or slope changes gradually.
How do I control the y intercept- the time delay in getting to the collision location due to the application of brakes? Ideally this would be a value that can be input into the equation as simply as "a" and "d" have been.
My graphing machine ideally will take 3 inputs: the speed of approach, the time of brake application, and the time delay necessary to avoid the collision, and it will produce the output of the necessary deceleration.