Tarzan is often portrayed as swinging through the jungle from vine to vine. Each of Tarzan's vines are 30 meters long and Tarzan catches each vine when it is at rest 15 degrees to the left of the (downwards) vertical, swings on it until it stops at an angle of 15 degrees to the right of vertical, and then grabs the next vine at rest and repeats the process.
What is Tarzan's average horizontal speed in m/s?
Details and assumptions
You may take g to be 9.8 m/s^2.
You may treat Tarzan as a simple pendulum and use the small angle approximation.
2.82
how do we solve it
?
To calculate Tarzan's average horizontal speed, we need to consider the period of each swing and the distance traveled during each swing.
First, let's calculate the period of Tarzan's swinging motion. We can use the formula for the period of a simple pendulum:
T = 2π√(L/g)
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
Given that the length of Tarzan's vine is 30 meters and g is 9.8 m/s^2, we can substitute these values into the formula:
T = 2π√(30/9.8)
T ≈ 6.078 seconds
Next, we need to calculate the distance Tarzan travels during each swing. Since Tarzan starts swinging from 15 degrees to the left of vertical and ends at 15 degrees to the right of vertical, the total angular displacement is 30 degrees (15 degrees to the left + 15 degrees to the right).
We can use the small angle approximation, which states that for small angles, the arc length is approximately equal to the radius multiplied by the angle in radians. Since an angle of 30 degrees is relatively small, we can apply this approximation.
The arc length can be calculated using the formula:
S = Lθ
where S is the arc length, L is the length of the vine, and θ is the angular displacement in radians.
Converting 30 degrees to radians:
θ = (30 * π) / 180
θ ≈ 0.523 radians
Substituting the values into the formula:
S = 30 * 0.523
S ≈ 15.7 meters
Now that we know the period (T) and the distance traveled during each swing (S), we can calculate the average horizontal speed (v) using the formula:
v = S/T
Substituting the values we calculated:
v = 15.7 / 6.078
v ≈ 2.58 m/s
Therefore, Tarzan's average horizontal speed is approximately 2.58 m/s.