A train is traveling at the rate 12 mi/h on a curve of radius 3000 ft. Through what angle has it turned in 1 min ?

To determine the angle through which a train traveling at a rate of 12 mi/h on a curve of radius 3000 ft has turned in 1 minute, we can use the formula for angular displacement:

\[
\theta = \frac{s}{r}
\]

where:
- \(\theta\) is the angular displacement in radians,
- \(s\) is the arc length traveled, and
- \(r\) is the radius of the curve.

### Step 1: Convert the Train's Speed to Feet per Minute

The speed of the train is given as 12 mi/h. First, let's convert this to feet per minute.

\[
12 \text{ mi/h} = 12 \times 5280 \text{ ft/h} = 63360 \text{ ft/h}
\]

Now, convert feet per hour to feet per minute:

\[
\text{Speed in ft/min} = \frac{63360 \text{ ft/h}}{60 \text{ min/h}} = 1056 \text{ ft/min}
\]

### Step 2: Calculate the Distance Traveled in 1 Minute

The distance \(s\) traveled by the train in 1 minute is the speed in feet per minute:

\[
s = 1056 \text{ ft}
\]

### Step 3: Calculate the Angular Displacement

Now, use the formula for angular displacement:

\[
\theta = \frac{s}{r} = \frac{1056 \text{ ft}}{3000 \text{ ft}}
\]

\[
\theta = 0.352 \text{ radians}
\]

### Step 4: Convert the Angle to Degrees

To convert radians to degrees, use the conversion factor \(1 \text{ rad} = \frac{180}{\pi} \text{ degrees}\):

\[
\theta = 0.352 \times \frac{180}{\pi} \approx 20.15^\circ
\]

### Conclusion

The train has turned through an angle of approximately **0.352 radians** or **20.15 degrees** in 1 minute.

Would you like more details or have any questions?

### Related Questions:
1. What would be the angle turned in 2 minutes?
2. How would the angle change if the train's speed were doubled?
3. How is angular velocity related to linear velocity in circular motion?
4. What is the formula for centripetal acceleration in circular motion?
5. How would the radius of the curve affect the angular displacement?

### Tip:
Remember that angular displacement is directly proportional to the arc length traveled and inversely proportional to the radius of the path!

Discover how to calculate the angle through which a train turns on a curve with a radius of 3000 ft, traveling at 12 mi/h. Learn the angular displacement formula and convert radians to degrees. Ideal for advanced high school students studying circular motion.

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