Olympic diver Matthew Mitcham springs upward from a diving board that is 3.20 m above the water. He enters the water at a 73.0 degree angle with respect to the water surface, at a speed of 9.36 m/s.

Determine the magnitude of his initial velocity.
Determine the direction of his initial velocity, in terms of degrees relative to horizontal.
Determine his maximum height above the water.

To determine the magnitude of Matthew Mitcham's initial velocity, we can use the components of his velocity in the x and y directions.

Given:
Displacement in x-direction (horizontal) = 0 m (since he springs upward)
Displacement in y-direction (vertical) = -3.20 m (height of diving board above water)
Angle with respect to water surface = 73.0 degrees
Speed = 9.36 m/s

Using trigonometry, we can find the components of the initial velocity:

velocity in x-direction (Vx) = speed * cos(angle)
Vx = 9.36 m/s * cos(73.0 degrees)

velocity in y-direction (Vy) = speed * sin(angle)
Vy = 9.36 m/s * sin(73.0 degrees)

To find the magnitude of initial velocity (V):

V = sqrt(Vx^2 + Vy^2)

Now, let's calculate V:

V = sqrt((9.36 m/s * cos(73.0 degrees))^2 + (9.36 m/s * sin(73.0 degrees))^2)

Using a calculator, we can find that V ≈ 9.50 m/s.

To determine the direction of the initial velocity in terms of degrees relative to horizontal, we need to find the angle (θ) that the initial velocity makes with the horizontal axis.

θ = arctan(Vy / Vx)

Now, let's calculate θ:

θ = arctan((9.36 m/s * sin(73.0 degrees)) / (9.36 m/s * cos(73.0 degrees)))

Again, using a calculator, we can find that θ ≈ 73.0 degrees.

Therefore, the magnitude of Matthew Mitcham's initial velocity is approximately 9.50 m/s, and his initial velocity makes an angle of approximately 73.0 degrees relative to horizontal.

To determine his maximum height above the water, we can use the kinematic equation:

Vy² = V0y² + 2 * a * Δy

Given:
Vy = 0 m/s (at maximum height, the vertical velocity is 0 since he changes direction)
a = -9.8 m/s² (acceleration due to gravity)
Δy = maximum height above the water

Plugging in the values, the equation becomes:

0 = (9.36 m/s * sin(73.0 degrees))² + 2 * (-9.8 m/s²) * Δy

Using this equation, we can solve for Δy, which will give us the maximum height above the water.

0 = (9.36 m/s * sin(73.0 degrees))² + 2 * (-9.8 m/s²) * Δy
0 = (9.36 m/s)^2 * sin²(73.0 degrees) - 19.6 m/s² * Δy

Δy ≈ (9.36 m/s * sin(73.0 degrees))² / (19.6 m/s²)

Again, using a calculator, we can find that Δy ≈ 1.13 m.

Therefore, Matthew Mitcham's maximum height above the water is approximately 1.13 m.

To find the magnitude of Matthew Mitcham's initial velocity, we can use the concept of projectile motion. We need to split the initial velocity into horizontal and vertical components.

1. The horizontal component of the initial velocity remains constant throughout the motion because there are no horizontal forces acting on the diver. Therefore, the horizontal component of the initial velocity is simply the initial speed, which is 9.36 m/s.

2. The vertical component of the initial velocity can be found using the trigonometric relationship between the angle and the components of the velocity. The vertical component can be calculated using the formula: vertical component = initial speed * sin(angle). Plugging in the given values, we get:
Vertical component = 9.36 m/s * sin(73°) ≈ 8.96 m/s.

3. To find the magnitude of the initial velocity, we can use the Pythagorean theorem. The magnitude (or total) of the initial velocity can be calculated using the formula: magnitude = √(horizontal component^2 + vertical component^2). Plugging in the values we found, we get:
Magnitude = √(9.36^2 + 8.96^2) ≈ 12.79 m/s.

Therefore, the magnitude of Matthew Mitcham's initial velocity is approximately 12.79 m/s.

To determine the direction of his initial velocity in terms of degrees relative to horizontal, we can use trigonometry again.

4. The direction can be calculated using the inverse tangent (arctan) function. The formula is: direction = arctan(vertical component / horizontal component). Plugging in the values, we get:
Direction = arctan(8.96 m/s / 9.36 m/s) ≈ 41.2°.

Therefore, the direction of Matthew Mitcham's initial velocity, relative to the horizontal, is approximately 41.2°.

To determine his maximum height above the water, we can use the concept of projectile motion and the kinematic equations.

5. The maximum height is reached when the vertical component of the velocity becomes zero. To calculate the time it takes to reach this point, we can use the formula: time = vertical component / acceleration, where acceleration is the acceleration due to gravity (-9.8 m/s^2). Plugging in the values, we get:
Time = 8.96 m/s / (-9.8 m/s^2) ≈ -0.915 s. Note that the negative sign indicates the direction of the acceleration.

6. Now that we have the time it takes to reach the maximum height, we can calculate the maximum height using the formula: maximum height = vertical component * time + (1/2) * acceleration * time^2. Plugging in the values, we get:
Maximum height = 8.96 m/s * (-0.915 s) + (1/2) * (-9.8 m/s^2) * (-0.915 s)^2 ≈ 3.59 m.

Therefore, Matthew Mitcham's maximum height above the water is approximately 3.59 meters.