A diver springs upward from a board that is three meters above the water. At the instant she contacts the water her speed is 9.10 m/s and her body makes an angle of 77.0° with respect to the horizontal surface of the water. Determine her initial velocity, both magnitude and direction.
The angle that her body makes with the water is not what matters here. What matters is is the angle that her center-of-mass trajectory makes with the water. Perhaps that is what they mean.
To determine the diver's initial velocity, we need to break down the given information into its vertical and horizontal components.
First, let's analyze the vertical motion. We know that the diver starts from rest at a height of 3 meters above the water. At the instant she contacts the water, her speed is 9.10 m/s, and her body makes an angle of 77.0° with respect to the horizontal surface of the water.
The vertical component of the initial velocity can be found using the following equation:
v_y = v * sinθ
where v_y is the vertical component of the initial velocity, v is the speed at contact (9.10 m/s), and θ is the angle with respect to the horizontal (77.0°).
Substituting the values into the equation, we get:
v_y = 9.10 m/s * sin(77.0°)
Next, let's find the horizontal component of the initial velocity. The horizontal component of the initial velocity remains constant throughout the motion. Hence, we can determine it using the following equation:
v_x = v * cosθ
where v_x is the horizontal component of the initial velocity, v is the speed at contact (9.10 m/s), and θ is the angle with respect to the horizontal (77.0°).
Substituting the values into the equation, we get:
v_x = 9.10 m/s * cos(77.0°)
Now that we have both the vertical and horizontal components of the initial velocity, we can determine the magnitude and direction of the initial velocity.
Magnitude (initial speed):
The magnitude of the initial velocity is given by the magnitude of the vector formed by the horizontal and vertical components:
|v| = √(v_x² + v_y²)
Direction (angle with respect to horizontal):
The direction of the initial velocity can be found using the following equation:
θ_initial = arctan(v_y / v_x)
Substituting the values into the equations, we can calculate the magnitude and direction of the initial velocity.