Let's solve the problem step by step: --- ## **Given Data** - **Shaft depth (\( \))**: 1, m - **Acceleration (\( \))**: 1 m/s² - **Braking (\( a \))**: 1 m/s² - **Maximum speed (\( v_{max} \))**: 18 m/s - **Loading, dumping, and creep time (\( t_{load} \))**: 50 s - **Payload per skip (\( Q \))**: 15 t - **Availability (\( A \))**: 95% --- ## **Step 1: Calculate Acceleration & Braking Distance and Time** ### **Time to accelerate to full speed:** \[ t_{accel} = \frac{v_{max}}{a} = \frac{18}{1} = 18 \text{ s} \] ### **Distance covered during acceleration:** \[ s_{accel} = \frac{1}{2} a t_{accel}^2 = \frac{1}{2} \times 1 \times (18)^2 = 162 \text{ m} \] ### **Braking is identical (1 m/s², to stop from 18 m/s):** \[ t_{brake} = \text{ s} \] \[ s_{brake} = 162 \text{ m} \] ### **Total distance at full speed:** \[ s_{full} = d - (s_{accel} + s_{brake}) = 120 - (162 + 162) = 876 \text{ m} \] ### **Time at full speed:** \[ t_{full} = \frac{s_{full}}{v_{max}} = \frac{876}{18} \approx 48.67 \text{ s} \] --- ## **Step 2: Total One-way Trip Time** \[ t_{one-way} = t_{accel} + t_{full} + t_{brake} \] \[ t_{one-way} = 18 + 48.67 + 18 = 84.67 \text{ s} \] --- ## **Step 3: Total Cycle Time** Each cycle consists of: - Down trip (empty): Same as up trip (loaded) - Loading, dumping, creep: 50 s \[ t_{cycle} = 2 \times t_{one-way} + t_{load} \] \[ t_{cycle} = 2 \times 84.67 + 50 = 169.34 + 50 = 219.34 \text{ s} \] --- ## **Step 4: Number of Cycles per Hour (with 95% availability)** \[ \text{Effective seconds per hour} = 360 \times .95 = 342 \text{ s} \] \[ \text{Number of cycles per hour} = \frac{342}{219.34} \approx 15.59 \] --- ## **Step 5: Calculate Production per Hour** \[ \text{Production per hour} = \text{Number of cycles per hour} \times \text{Payload per skip} \] \[ \text{Production per hour} = 15.59 \times 15 = 233.85 \text{ t/hr} \] --- ## **Final Answer** \[ \boxed{233.9\ \text{metric t/hr}} \] --- ### **Summary Table** | Step | Value | |---------------------------|---------------| | Shaft depth | 1,200 m | | Acceleration/braking | 1 m/s² | | Full speed | 18 m/s | | One-way travel time | 84.67 s | | Cycle time (total) | 219.34 s | | Available time per hour | 3,420 s | | Cycles per hour | 15.59 | | Payload per skip | 15 t | | **Production per hour** | **233.9 t/hr** | --- If you need further clarifications, let me know!