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.