Olympic diver Matthew Mitcham springs upward from a diving board that is 3.00 m above the water. He enters the water at a 79.0 degree angle with respect to the water surface, at a speed of 9.01 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.

First we must break it up into its individual x and y directions.

velocity in y direction =9.01*cos(11)= 8.8445 m/s downwards
velocity in x direciton =9.01*cos(79)=
1.71918 m/s to the right.

Given Vfy(velocity final y direction) we can find max height by using V=Vi+At acceleration being -9.81m/s^2.
using that formula we get 3.9869 meters as the max height reached.

As the diver starts 3 meters above the water 3.9869m-3m=.9869 m we can use this to find initial velocity in y direction using equation (vf)^2-(vi)^2=2a(xf-xi)
plugging this in given vfy = 0 at the top of the curve we get Viy= 4.4003 m/s
Because there is no horizontal acceleration Vix=1.71918 m/s (found at beginning). To get the magnitude of this velocity we use the Pythagorean thrm. 4.4003^2+1.7198^2=(4.724)^2
initial magnitude=4.724 m/s

Finally to determine direction we use inverse cosine of Vx/Vmagnitude to get a final answer of 68 degrees with respect to the horizontal.

equations used:
v²-v₀²=2a(x-x₀) v = v₀ + at
a^2+b^2=c^2 and inverse cosine

To determine the magnitude of Matthew Mitcham's initial velocity, we can use the fact that the vertical component of his velocity is equal to his initial speed, and the horizontal component is zero.

Given:
Height of the diving board, h = 3.00 m
Angle with respect to the water surface, θ = 79.0 degrees
Speed at entry, v = 9.01 m/s

1. Magnitude of the Initial Velocity:
Since the vertical component of the velocity is equal to the initial speed, we can use trigonometry to find the magnitude of the initial velocity (v₀). The vertical component can be calculated using the sine function:

vertical component: vₓ = v * sin(θ)

Therefore, we have:

Vertical component (vₓ) = v * sin(θ) = 9.01 m/s * sin(79.0°) ≈ 8.92 m/s

Since the horizontal component is zero, the magnitude of the initial velocity (v₀) can be equated to the vertical component:

Magnitude of Initial Velocity (v₀) = Vertical component (vₓ) ≈ 8.92 m/s

2. Direction of Initial Velocity:
To determine the direction of the initial velocity, we can use the cosine function to calculate the horizontal component of the velocity:

horizontal component: vᵧ = v * cos(θ)

Since the horizontal component is zero, the direction of the initial velocity is 90 degrees relative to horizontal.

3. Maximum Height above the Water:
The maximum height above the water can be determined using the kinematic equation for vertical motion:

h_max = (vₓ²) / (2 * g)

where g is the acceleration due to gravity (9.8 m/s²).

Substituting the known values, we have:

Maximum Height (h_max) = (vₓ²) / (2 * g) = (8.92 m/s)² / (2 * 9.8 m/s²) ≈ 4.052 m

Therefore, Matthew Mitcham's maximum height above the water is approximately 4.052 meters.