There is a 1500 mile race course. I MUST have a top speed of 36 meters/s (meaning I cannot go over 36). 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 you could get to 800 meters, we need to consider the acceleration and velocity limits.
Let's break down the problem into different phases to determine the optimal strategy.
Phase 1: Acceleration from 0 to maximum velocity (36 m/s)
In this phase, you want to accelerate to your maximum velocity as quickly as possible without exceeding it. The maximum forward acceleration is given as 4 m/s^2, so you can reach a velocity of 36 m/s in:
t1 = (36 m/s) / (4 m/s^2) = 9 seconds
During this phase, you will cover a distance of:
d1 = (1/2) * (4 m/s^2) * (t1^2) = (1/8) * (81) = 10.125 meters
Phase 2: Deceleration from maximum velocity (36 m/s) to 0
Now, you need to decelerate to a stop at the 800-meter mark. The maximum backward (deceleration) is given as 6 m/s^2. Since the distance you need to cover in this phase is 200 meters, let's calculate the time (t2) required to decelerate to 0 velocity:
t2 = (36 m/s) / (6 m/s^2) = 6 seconds
During this phase, you will cover a distance of:
d2 = (1/2) * (6 m/s^2) * (t2^2) = (1/2) * (36) = 18 meters
Phase 3: Constant velocity
After deceleration, you will be at the 800-meter mark, and you need to maintain a constant velocity of 0 m/s until the race is completed. Since your velocity is 0, there is no time or distance associated with this phase.
Equation for Time, Position, and Velocity:
The total time required to complete the race will be the sum of the times for each phase:
Total time = t1 + t2 = 9 + 6 = 15 seconds
The total distance covered will be the sum of the distances for each phase:
Distance traveled = d1 + d2 + 200m = 10.125 + 18 + 200 = 228.125 meters
The velocity varies during the different phases:
- In Phase 1: The velocity increases from 0 to 36 m/s within 9 seconds.
- In Phase 2: The velocity decreases from 36 m/s to 0 within 6 seconds.
- In Phase 3: The velocity remains at 0 m/s.
Therefore, the equation for the position, time, and velocity of your vehicle during the race can be written as a piecewise function:
Position (x) as a function of time (t):
x(t) = { 0.5 * 4 * t^2 if 0 <= t <= 9
{ 10.125 + 36 * (t-9) if 9 < t <= 15
{ 228.125 if t > 15
Velocity (v) as a function of time (t):
v(t) = { 4 * t if 0 <= t <= 9
{ 36 - 6 * (t-9) if 9 < t <= 15
{ 0 if t > 15
Please note that these equations assume instant and perfect control of acceleration and deceleration, which may not be physically achievable in practice.