A swimmer of mass m is at a swimming pool. They climb the steps to the diving board 4.0 m above the water. They jump off the board, which enables them to rise a further 0.7 m before they begin their descent into the pool. At what speed do they enter the water?
You may assume that the acceleration due to gravity, g, is 9.8 m s−2
Show your working step by step, including the correct formulas and equations.
m*g*h = 1/2*m*v^2
where m is the mass; g is the acceleration due to gravity; v is the speed.
At the top of his dive, the swimmers energy is entirely potential energy = m*g*h;
When he enters the pool, the potential energy is converted to kinetic energy: 1/2*m*v^2
v = = (2*g*h)^0.5
9.6
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 = √(2 * g * h)
Now we can plug in the given values and calculate the speed:
velocity = √(2 * 9.8 * 4.7)
velocity = √(92.12)
velocity ≈ 9.60 m/s
Therefore, the swimmer enters the water with a speed of approximately 9.60 m/s.