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.
4.7 = 4.9t^2
t = .979 sec
v = 9.8t = 9.8*.979 = 9.6 m/s
To solve this problem and find the speed at which the swimmer enters the water, we can use the principles of conservation of mechanical energy.
Step 1: Calculate the potential energy at the diving board.
The potential energy at the diving board is given by the formula:
PE = mgh
where PE is the potential energy, m is the mass of the swimmer, g is the acceleration due to gravity, and h is the height of the diving board.
In this case, the height is 4.0 meters, so we have:
PE_board = mgh = m * 9.8 * 4.0 = 39.2m Joules.
Step 2: Calculate the potential energy at the highest point of ascent.
When the swimmer rises 0.7 meters above the diving board, their potential energy at this point can be calculated using the same formula:
PE_ascent = mgh
where h is the height of ascent, which is 0.7 meters.
So, PE_ascent = m * 9.8 * 0.7 = 6.86m Joules.
Step 3: Calculate the kinetic energy at the highest point of ascent.
At the highest point of ascent, the swimmer has no potential energy left. Therefore, all the energy is in the form of kinetic energy.
Kinetic energy is given by the formula:
KE = 1/2 * m * v^2
where KE is the kinetic energy and v is the velocity.
So, using the known value of PE_ascent, we can equate the kinetic energy at the highest point to the potential energy at the highest point:
KE_ascent = PE_ascent
1/2 * m * v_ascent^2 = 6.86m
Dividing both sides by m, we have:
1/2 * v_ascent^2 = 6.86
v_ascent^2 = 13.72
v_ascent = √13.72
v_ascent ≈ 3.70 m/s
Step 4: Calculate the final velocity at the water surface.
Since the swimmer is descending from the point of highest ascent to the water surface, they will gain additional potential energy, which will then be converted to kinetic energy by the time they reach the water.
The potential energy at the water's surface is given by:
PE_descent = mgh
where h is the descent height, which is equal to 0.7 meters.
So, PE_descent = m * 9.8 * 0.7 = 6.86m Joules.
And, using the formula for kinetic energy stated earlier, we can equate the potential energy at the water's surface to the kinetic energy of the swimmer at that point:
PE_descent = KE_descent
6.86m = 1/2 * m * v_descent^2
Dividing both sides by m, we have:
1/2 * v_descent^2 = 6.86
v_descent^2 = 13.72
v_descent = √13.72
v_descent ≈ 3.70 m/s
Step 5: Calculate the total velocity at the water surface.
Since velocity is a vector quantity, we need to consider its direction. The swimmer jumps off the diving board and rises to the highest point of ascent with a positive velocity, and then descends to the water with a negative velocity.
Therefore, the total velocity will be the difference between the ascent velocity and the descent velocity:
v_total = v_ascent - v_descent
v_total = 3.70 m/s - (-3.70 m/s)
v_total = 3.70 m/s + 3.70 m/s
v_total = 7.40 m/s
Thus, the swimmer enters the water with a speed of 7.40 m/s.