A diver springs upward from a board that is 3.00 m above the water. At the instant she contacts the water her speed is 8.90 m/s and her body makes an angle of 75.0 degrees with respect to the horizontal surface of the water. Determine her initial velocity both magnitude and direction.
To determine the initial velocity of the diver, we can use the concepts of vertical and horizontal motion. Let's break it down step by step:
Step 1: Identify the given information
- Vertical distance (h) = 3.00 m
- Final speed (vf) = 8.90 m/s
- Angle with respect to the horizontal (θ) = 75.0 degrees
Step 2: Determine the vertical component of the final velocity
The vertical component of the velocity (vfy) can be found using the formula:
vfy = vfsin(θ)
Substituting the values, we get:
vfy = 8.90 m/s * sin(75.0°)
Calculating this, we find:
vfy = 8.90 m/s * 0.9659
vfy ≈ 8.58 m/s
Step 3: Determine the time taken to reach the water
To find the time taken (t) from the top of the dive board to the water's surface, we can use the vertical motion equation:
h = vyi * t + (1/2) * g * t^2
Since the diver starts from rest at the top, the initial vertical velocity (vyi) is 0 m/s. Also, we know that acceleration due to gravity (g) is approximately 9.8 m/s^2.
Substituting the values, we have:
3.00 m = 0 * t + (1/2) * 9.8 m/s^2 * t^2
Simplifying the equation, we get:
4.9 t^2 = 3
Solving for t gives us:
t^2 = 3 / 4.9
t ≈ √(0.6122)
t ≈ 0.782 s
Step 4: Determine the initial velocity components
To find the initial velocity (vxi and vyi), we can use the equations:
vxi = vfx
vyi = vfy - g * t
Since there is no horizontal acceleration, the final horizontal velocity (vfx) is equal to the initial horizontal velocity (vxi). Using the angle (θ) and the final speed (vf), we can calculate vfx and vfy.
vfx = vfsin(θ)
vfy = vfcos(θ)
Substituting the values, we get:
vxi = vfsin(θ) ≈ 8.90 m/s * sin(75.0°)
vyi = vfcos(θ) ≈ 8.90 m/s * cos(75.0°)
Calculating these values, we find:
vxi ≈ 8.90 m/s * 0.9659
vxi ≈ 8.59 m/s
vyi ≈ 8.90 m/s * 0.2588
vyi ≈ 2.30 m/s
Step 5: Calculate the magnitude and direction of the initial velocity
The magnitude of the initial velocity (vi) can be found using the Pythagorean theorem:
vi = √(vxi^2 + vyi^2)
Substituting the values, we have:
vi = √(8.59 m/s)^2 + (2.30 m/s)^2
vi ≈ √73.92 m^2/s^2
vi ≈ 8.60 m/s
The direction of the initial velocity (θi) can be found using the inverse tangent function:
θi = atan(vyi / vxi)
Substituting the values, we get:
θi = atan(2.30 m/s / 8.59 m/s)
Calculating this, we find:
θi ≈ atan(0.2680)
θi ≈ 15.1 degrees
Therefore, the initial velocity of the diver is approximately 8.60 m/s at an angle of 15.1 degrees with respect to the horizontal surface of the water.
To determine the diver's initial velocity, we need to break it down into its horizontal and vertical components.
First, we can find the horizontal component of the velocity. Since there is no horizontal acceleration, the initial horizontal velocity remains constant throughout the motion. Therefore, the horizontal component of the velocity at any point is the same as the initial horizontal velocity.
Next, we can find the vertical component of the velocity. We know that the speed of the diver just before hitting the water is 8.90 m/s, and the angle of her body with respect to the horizontal surface of the water is 75.0 degrees. Using trigonometry, we can find the vertical component of the diving velocity.
Vertical component of velocity = speed × sin(angle)
= 8.90 m/s × sin(75.0 degrees)
Now, let's calculate the vertical component of the velocity:
Vertical component of velocity = 8.90 m/s × sin(75.0 degrees)
= 8.90 m/s × 0.9659
Now, plug this value into your calculator to find the vertical component of the velocity.
Vertical component of velocity ≈ 8.10 m/s
Since there is no initial vertical velocity (the diver springs upward), the initial vertical component of the velocity is equal to the vertical component of the velocity just before hitting the water. Therefore, the initial vertical velocity is 8.10 m/s.
Finally, we can use the initial horizontal and vertical velocities to find the magnitude and direction of the initial velocity.
To find the magnitude of the initial velocity, we can use the Pythagorean theorem:
Initial velocity = √(horizontal component^2 + vertical component^2)
= √(constant value^2 + 8.10 m/s^2)
Now, calculate the magnitude of the initial velocity:
Initial velocity ≈ √(constant value^2 + 8.10 m/s^2)
As we don't have the constant horizontal value, we cannot determine the exact magnitude of the initial velocity without additional information.
To find the direction of the initial velocity, we can use the tangent function:
Direction = atan(vertical component / horizontal component)
= atan(8.10 m/s / constant value)
Again, without the horizontal component or additional information, we cannot determine the exact direction of the initial velocity.
In summary, the initial velocity of the diver consists of a horizontal component (which stays constant) and a vertical component of 8.10 m/s. The magnitude and direction of the initial velocity cannot be determined without additional information.