You like to swim at a nearby lake. On one side of the lake is a cliff, and the top of the cliff is 69 m above the surface of the lake. You dive of the cliff doing somersaults so as to have an angular speed of 5.5 rev/s. How many revolutions do you make before you hit the water? Assume your initial centre of mass velocity is horizontal and you begin from a standing position.
so, the only thing to calculate is the time to fall.
Multiply that by 5.5 to get the revs.
Hmmm. How about the old s = 1/2 at^2
Sloppy language. The center of mass is a point, and cannot be horizontal. Assume the velocity is horizontal, so there is no initial vertical component.
And watch out for those rocks at the bottom of the cliff!!! That's where the horizontal velocity comes in. :-)
And, the water better be shallow. At 5.5rev/s you'll be too dizzy to swim when you hit!
To solve this problem, we need to consider the motion of the diver as they fall from the cliff and enter the water. We can break down the problem into two parts: the vertical motion of the diver and the rotational motion of the somersaults.
First, let's consider the vertical motion. We can use the equation of motion to determine the time it takes for the diver to reach the water's surface. The equation is:
h = v₀t + (1/2)gt²
Where:
h is the height of the cliff (69m),
v₀ is the initial velocity (0 m/s since the diver starts from a standing position),
g is the acceleration due to gravity (-9.8 m/s²),
and t is the time.
Since the diver starts from rest, the equation simplifies to:
h = (1/2)gt²
Plugging in the given values, we have:
69 = (1/2)(-9.8)t²
Rearranging the equation to solve for t:
t² = (69 * 2) / 9.8
t² = 14.08
Taking the square root of both sides:
t ≈ √14.08
t ≈ 3.75 seconds
So, it will take approximately 3.75 seconds for the diver to reach the water.
Next, let's consider the rotational motion of the somersaults. The angular speed is given as 5.5 rev/s, which means the diver completes 5.5 revolutions every second.
Since we know the time it takes for the diver to reach the water (3.75 seconds), we can calculate the total number of revolutions they make during this time:
Number of revolutions = (angular speed) * (time)
Number of revolutions = 5.5 rev/s * 3.75 s
Number of revolutions ≈ 20.63
Therefore, the diver makes approximately 20.63 revolutions before hitting the water.