Projectile Weaving Machine Speed Calculation

This solution details the steps to calculate the maximum loom speed in Picks Per Minute (PPM) for a projectile weaving machine, using the provided distance and average velocity.

1. Convert Units and Identify Given Data

Ensure consistent units for calculation. Convert the distance from centimeters (cm) to meters (m).

• Distance ($d$): $250 \text{ cm} = 2.5 \text{ m}$
• Average Velocity ($v$): $25 \text{ m/s}$
• Condition: Motion time is half the total loom cycle time.

2. Calculate Time of Projectile Motion

Use the relationship between distance, velocity, and time ($t = d/v$).

$ t_{\text{motion}} = \frac{d}{v} $

Plugging in the values:

$ t_{\text{motion}} = \frac{2.5 \text{ m}}{25 \text{ m/s}} = 0.1 \text{ s} $

3. Determine Total Loom Cycle Time

The problem states that the time the projectile is in motion ($t_{\text{motion}}$) is half the total loom cycle time ($T_{\text{cycle}}$).

$ T_{\text{cycle}} = 2 \times t_{\text{motion}} $

$ T_{\text{cycle}} = 2 \times 0.1 \text{ s} = 0.2 \text{ s} $

4. Calculate Maximum Loom Speed (Picks Per Minute)

Loom speed is measured in Picks Per Minute (PPM). To find PPM, convert the cycle time from seconds to minutes or use the formula relating seconds per minute to the cycle time in seconds.

$ \text{PPM} = \frac{\text{Seconds per Minute}}{T_{\text{cycle}} (\text{in seconds})} $

$ \text{PPM} = \frac{60 \text{ s/min}}{0.2 \text{ s}} = 300 \text{ picks/min} $

Therefore, the maximum loom speed is 300 PPM.