To determine Tarzan's average horizontal speed while swinging from vine to vine, we can model his motion as that of a simple pendulum. The key here is to understand how the pendulum behaves and how we can calculate the time it takes for Tarzan to swing from one side to the other, as well as the distance he travels horizontally during that time.
Understanding the Pendulum Motion
In this scenario, Tarzan swings from an angle of 15 degrees to the left of vertical to 15 degrees to the right of vertical. The total angle of swing is therefore 30 degrees. We can use the small angle approximation, which simplifies our calculations. This approximation states that for small angles (in radians), the sine of the angle is approximately equal to the angle itself.
Calculating the Length of the Swing
First, we need to find the length of the arc that Tarzan travels while swinging. The length of the vine is given as 30 meters. The angle in radians for 15 degrees is:
- 15 degrees = 15 × (π / 180) = π / 12 radians
Using the formula for the arc length \( s \) of a circle, which is \( s = r \theta \) (where \( r \) is the radius and \( \theta \) is the angle in radians), we can calculate the distance Tarzan swings:
- For one side: \( s = 30 \times \frac{\pi}{12} = \frac{30\pi}{12} = 7.85 \text{ meters} \)
Since he swings from left to right, the total distance for one complete swing (left to right) is:
- Total distance = \( 2 \times 7.85 = 15.7 \text{ meters} \)
Determining the Period of the Swing
The period \( T \) of a simple pendulum can be approximated using the formula:
- \( T = 2\pi \sqrt{\frac{L}{g}} \)
Where \( L \) is the length of the pendulum (30 meters) and \( g \) is the acceleration due to gravity (9.8 m/s²). Plugging in the values:
- \( T = 2\pi \sqrt{\frac{30}{9.8}} \approx 2\pi \sqrt{3.06} \approx 2\pi \times 1.75 \approx 11.0 \text{ seconds} \)
Calculating Average Horizontal Speed
Now that we have the total distance Tarzan swings and the time it takes for one complete swing, we can calculate his average horizontal speed \( v \). The average speed is given by the formula:
- \( v = \frac{\text{Total distance}}{\text{Total time}} \)
Substituting the values we have:
- \( v = \frac{15.7 \text{ meters}}{11.0 \text{ seconds}} \approx 1.43 \text{ m/s} \)
Final Thoughts
Thus, Tarzan's average horizontal speed while swinging from vine to vine is approximately 1.43 m/s. This analysis not only illustrates the principles of pendulum motion but also emphasizes the beauty of physics in understanding everyday actions, even those of a fictional character like Tarzan!
