A variable rate liquid nitrogen applicator travels across a field at $v = 7.2\text{ km/h}$. The boom control system receives prescription application commands from the onboard GIS controller. Due to hydraulic valve response and chemical transport through delivery lines, there is a physical actuator lag time of $t_{\text{lag}} = 1.25\text{ s}$. To ensure the prescribed fertilizer rate is deposited precisely on the target prescription zone, the look-ahead distance in $\text{m}$ ahead of the boom must be set to ________ (round off to 1 decimal place).
Answer Key:
2.5
Step-by-Step Solution & Explanation
```
Look-Ahead Distance Compensation:
Tractor Position =======( d_look )======> Target Grid Zone
v = 2.0 m/s
```
**Step-by-step Calculation:**
1. Forward travel speed in $\text{m/s}$:
$v = \frac{7.2 \times 1000}{3600} = 2.0\text{ m/s}$
2. Look-ahead distance ($d$):
To compensate for total system latency $t_{\text{lag}} = 1.25\text{ s}$, the controller must trigger actuator commands when the machine is at distance $d$ prior to crossing into the new prescription zone:
$d = v \times t_{\text{lag}} = 2.0\text{ m/s} \times 1.25\text{ s} = 2.50\text{ m}$