There is a 1500 mile race course. The max foward acceleration is 4 m/s squared and the max backward acceleration is 6 m/s squared.
I start at 600m and must go to the 800m mark (I must travel 200m). What is an equation for the fastest time that I could get to 800m? What is the position, time and velocity of my vehical?
To find the equation for the fastest time to travel from 600m to 800m, we can break down the problem into two parts: accelerating from 600m to some intermediate point, and then decelerating from that intermediate point to 800m.
Let's denote the position at the intermediate point as x (where 600m ≤ x ≤ 800m). The time taken to accelerate to this point can be found using the equation:
t1 = √((2 * (x - 600m)) / a1)
Where:
- t1 is the time taken to reach the intermediate point,
- x is the position at the intermediate point (in meters),
- 600m is the starting position,
- a1 is the maximum forward acceleration (4 m/s²).
Next, we need to consider the time taken to decelerate from the intermediate point to the target position of 800m. The time taken to decelerate can be calculated using the equation:
t2 = √((2 * (800m - x)) / a2)
Where:
- t2 is the time taken to decelerate from the intermediate point to the target position,
- a2 is the maximum backward acceleration (6 m/s²).
The total time taken to travel from 600m to 800m is then the sum of t1 and t2:
total time = t1 + t2
Now, let's calculate the intermediate point (x) and the corresponding time (t1) using the given values:
1. Solve for x:
t1 = √((2 * (x - 600m)) / 4 m/s²)
Square both sides: t1^2 = (2 * (x - 600m)) / 4 m/s²
Multiply both sides by 4 m/s²: (4 m/s²) * t1^2 = 2 * (x - 600m)
Divide both sides by 2: (2 m/s²) * t1^2 = x - 600m
Add 600m to both sides: x = (2 m/s²) * t1^2 + 600m
2. Substitute the known values of t1 and solve for x:
x = (2 m/s²) * t1^2 + 600m
x = (2 m/s²) * (√((2 * (x - 600m)) / 4 m/s²))^2 + 600m
Simplify: x = (1 m/s) * √(2 * (x - 600m)) + 600m
Subtract 600m from both sides: x - 600m = (1 m/s) * √(2 * (x - 600m))
Square both sides: (x - 600m)^2 = (1 m/s)^2 * (2 * (x - 600m))
Simplify: x^2 - 1200m * x + 360000m^2 = 2m^2 * x - 2400m^2 + 1200m * x
Rearrange: x^2 - 1200m * x + 360000m^2 - 2m^2 * x + 2400m^2 - 1200m * x = 0
Combine like terms: x^2 - 1200m * x - 2m^2 * x + 360000m^2 + 2400m^2 - 1200m * x = 0
Combine like terms: x^2 - 3600m * x + 600000m^2 = 0
This quadratic equation will help us find the values of x.