Suppose the diver vaults off the springboard, leaving it with an initial speed of 3.27 m/s upwards. Use energy conservation to find his speed when he strikes the water.
1.5
To find the speed of the diver when he strikes the water, we can use the principle of conservation of energy. According to this principle, the total mechanical energy of the system remains constant if no external forces act on it.
The total mechanical energy of the system can be expressed as the sum of the kinetic energy (KE) and the potential energy (PE):
E = KE + PE
At the start, when the diver leaves the springboard with an initial speed of 3.27 m/s upwards, he does not possess any potential energy, as he is at ground level. Therefore, the total mechanical energy is equal to the initial kinetic energy:
E = KE_initial
When the diver strikes the water, he loses all of his initial kinetic energy and gains potential energy due to being at a higher position in the water. Therefore, the total mechanical energy at this point is equal to the potential energy:
E = PE_final
Since mechanical energy is conserved, we can equate the initial kinetic energy to the final potential energy:
KE_initial = PE_final
The initial kinetic energy can be calculated using the formula:
KE_initial = (1/2)mv^2
where m is the mass of the diver and v is the initial speed.
To find the final speed when the diver strikes the water, we need to calculate the potential energy. The potential energy can be expressed as:
PE_final = mgh
where m is the mass of the diver, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the water above the diver when he strikes it.
Combining these equations, we have:
(1/2)mv^2 = mgh
Dividing both sides by m and cancelling out the mass, we get:
(1/2)v^2 = gh
Finally, we can solve for the final speed:
v^2 = 2gh
v = sqrt(2gh)
where v is the final speed of the diver when he strikes the water, g is the acceleration due to gravity, and h is the height of the water above the diver.
To find the diver's speed when he strikes the water using energy conservation, we need to consider the conservation of mechanical energy. The mechanical energy of the diver-springboard system is conserved as long as only conservative forces are involved.
The mechanical energy of the system before the diver jumps consists of two parts: potential energy (PE) and kinetic energy (KE).
1. Potential Energy (PE):
The potential energy of an object in a gravitational field is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s²), and h is the height above a reference point (usually the ground). Since the diver vaults off the springboard, the reference point is usually taken as the water's surface. Therefore, the potential energy before the jump is zero.
2. Kinetic Energy (KE):
The kinetic energy of an object is given by the equation KE = (1/2)mv², where m is the mass and v is the speed. Before the diver jumps, he has an initial speed of 3.27 m/s upwards. Since the diver travels upwards, we can consider this as the initial kinetic energy.
According to the conservation of mechanical energy, the total mechanical energy before and after the jump must remain the same.
E_initial = E_final
Since the initial potential energy is zero:
KE_initial = KE_final
Substituting the values:
(1/2)mv_initial² = (1/2)mv_final²
Simplifying the equation:
v_initial² = v_final²
Now, we can simply take the square root of both sides of the equation to determine the final velocity:
v_final = √v_initial²
v_final = √(3.27 m/s)²
v_final ≈ 3.27 m/s
Therefore, the diver's speed when he strikes the water is approximately 3.27 m/s.
They should have told you the springboard height above the water. For Olympic competition, it is 3.0 meters.
That height, the initial speed and conservation of energy should tell you what the water impact speed is.