You are on the right track, but you have made a couple of errors. One is a units mix-up and the other is changing from a force based to speed based calculation without working completely through the problem. Unfortunately you don't get to choose top speed and acceleration with a fixed motor, you trade one against the other.

6m/s is crazy fast for a line follower and you will find you will never see that with the constraints you have. You need to calculate from the constraints, which are your 'bots mass, the motor you have, and the incline on the course.

3.42 m/s

^{2} is the decceleration due to gravity your bot must overcome to move up a 20 degree slope, so to give it a little leeway (but mostly to make the maths easier

) call your minimum acceleration requirement 4m/s

^{2}. Since you have 2 motors you want each to deliver enough force for a 2m/s

^{2} acceleration.

Now convert required acceleration to force:

F=MA = 1kg x 2m/s

^{2} = 2N

And compare to the force your motors provide (running, not stalled):

Torque=20g.cm at motor = 20/3.75 = 5.33g(F) (thats grams force,

**not** Newtons

) at the wheel.

so convert g(F) to Newtons:

1N ~100g(F) so 5.33g(F) = 0.0533N

Desired gear ratio will be the ratio of force provided:required -

= 0.0533:2 = 1:37.5

**Now** you can take the gear ratio and calculate your speed:

Speed = ((motor_rpm/60) x pi x wheel_diameter) / gear_ratio = ((7800/60) x 3.14 x 0.075)/37.5 = 0.82m/s

Speed may seem low but this is not too bad for a line follower, and if you try to go much faster you run the risk of your 'bot not being able to drive up the incline, or track the line properly. You should get more points for a slow finish than a quick failure. If you want more performance try to remove some mass from your 'bot, halve your mass and you double your acceleration for a given motor.

N.B. Approximations used throughout and no allowance made for system inefficiency.