Racing Time Difference Formula

Question

I am trying to develop an android app for a friend that uses the gps to tell you how many seconds ahead or behind you are from your target speed vs your actual speed.

For example I could drive for 1 min at 10mph, with '60mph' selected on the app. The app would then show me '-300sec' because I would be 5min behind if I had been going 60mph for 1min instead.

The formula I have is:

((actualSpeed - targetSpeed) / actualSpeed) * Time

so for the example: ((10mph - 60mph)/10mph) * 60sec = -300sec

I thought my formula was fine until i tried a different example:

I could drive for 1 min at 60mph, with '30mph' selected on the app. The app should then show me '+60sec' because I would be 1min ahead if I had been going 30mph for 1min instead.

so for the example: ((60mph - 30mph)/60mph) * 60sec = +30sec

With this example I'm getting +30sec instead of +60sec

I'm thinking that this formula has been used a lot with racing to optimize time but I can't seem to get a formula that works for all inputs.

Thanks in advance.

Answer 1

Try $\displaystyle \left( 1-\frac{\text{target speed}}{\text{actual speed}} \right) \times \text{time}$

Answer 2

As you are travelling the same distance, $d_1=d_2=d$, where $\cdot_1$ indicates the target results and $\cdot_2$ indicates the actual results.

We know $d=s_1t_1=s_2t_2$, where $t_*$ is the time and $s_*$ is the speed. You are interested in $t_2-t_1$. The above equation can be solved for $t_1$ as $t_1=\dfrac{s_2}{s_1}t_2$. The difference in time is then $$t_2-\dfrac{s_2}{s_1}t_2=\left(1-\dfrac{s_2}{s_1}\right)t_2.$$ This is the forumula for the diffence in time where $s_1$ is your target speed, $s_2$ is your actual speed and $t_2$ is the actual time it took.