A diver springs upward from a board that is 4.10 m above the water. At the instant she contacts the water her speed is 11.9 m/s and her body makes an angle of 79.7 ° with respect to the horizontal surface of the water. Determine her initial velocity, both (a) magnitude and (b) direction.
To determine the initial velocity of the diver, we can use the principles of projectile motion and vector decomposition.
(a) To find the magnitude of the initial velocity, we need to break down the motion into vertical and horizontal components. Considering that the diver starts from rest horizontally and only gravity acts vertically, we can analyze these components to find the initial velocity.
Vertical Component:
The diver starts from the board with an initial vertical velocity of 0 m/s. Considering the vertical motion, we can use the following equation:
vf^2 = vi^2 + 2as
Where vf is the final vertical velocity (which is 11.9 m/s at the instant she contacts the water), vi is the initial vertical velocity, a is the acceleration due to gravity (-9.8 m/s^2), and s is the displacement vertically (which is -4.10 m, taking downward as negative).
Substituting the given values into the equation, we have:
(11.9 m/s)^2 = (vi)^2 + 2(-9.8 m/s^2)(-4.10 m)
Solving for vi, we get:
vi^2 = (11.9 m/s)^2 - 2(-9.8 m/s^2)(-4.10 m)
vi^2 = 141.61 m^2/s^2 - 80.392 m^2/s^2
vi^2 = 61.218 m^2/s^2
Taking the square root of both sides, we find:
vi = √(61.218 m^2/s^2)
vi ≈ 7.82 m/s
Therefore, the magnitude of the initial velocity is approximately 7.82 m/s.
(b) To determine the direction of the initial velocity, we can use trigonometry. The angle of 79.7° with respect to the horizontal surface is the angle of the resultant velocity vector. The initial velocity vector can be represented as the sum of its horizontal and vertical components.
The horizontal component of the initial velocity (vix) can be found using the equation:
vix = vi * cosθ
Where θ is the angle of 79.7°.
Substituting the given values into the equation, we have:
vix = 7.82 m/s * cos(79.7°)
vix ≈ 7.82 m/s * (-0.195)
vix ≈ -1.52 m/s
The negative sign indicates that the horizontal component of the initial velocity is in the opposite direction of the positive x-axis. Therefore, the initial velocity has a horizontal component of approximately -1.52 m/s.
Hence, the initial velocity's magnitude is 7.82 m/s, and its direction is in the negative x-axis with a horizontal component of approximately -1.52 m/s.