To find the speed at which the swimmer enters the water, we can use the principle of conservation of mechanical energy. The total mechanical energy of the swimmer consists of two parts - potential energy (due to their position above the water) and kinetic energy (due to their motion).
Step 1: Calculate the potential energy at the top of the diving board:
The potential energy formula is given by:
Potential energy = mass * gravitational acceleration * height
Since the swimmer is at rest, all their energy is in the form of potential energy. The height we consider here is the total height above the water, which is the height of the diving board plus the extra rise before descent:
Height = 4.0 m + 0.7 m = 4.7 m
So, the potential energy is:
Potential energy = m * g * h
Step 2: Calculate the kinetic energy at the surface of the water:
At the surface of the water, all the potential energy has been converted to kinetic energy, according to the principle of conservation of mechanical energy.
Kinetic energy = potential energy
0.5 * mass * velocity^2 = m * g * h
Step 3: Solve for velocity:
Rearranging the equation to solve for velocity gives:
velocity^2 = 2 * g * h
Taking the square root of both sides gives:
velocity = ā(2 * g * h)
Now we can plug in the given values and calculate the speed:
velocity = ā(2 * 9.8 * 4.7)
velocity = ā(92.12)
velocity ā 9.60 m/s
Therefore, the swimmer enters the water with a speed of approximately 9.60 m/s.