A diver springs upward from a board that is three meters above the water. At the instant she contacts the water her speed is 8.70 m/s and her body makes an angle of 74.0° with respect to the horizontal surface of the water. What is her initial velocity, both magnitude and direction?
To find the initial velocity of the diver, we can break down the initial velocity into horizontal and vertical components.
The horizontal component of the initial velocity remains constant throughout the motion because there are no horizontal forces acting on the diver. Therefore, the horizontal component of the initial velocity (Vx) is the same as the horizontal component of the final velocity which is given by:
Vx = Vf * cos(θ)
where Vf is the final velocity (8.70 m/s) and θ is the angle the body makes with respect to the horizontal surface of the water (74.0°).
Plugging in the values, we get:
Vx = 8.70 m/s * cos(74.0°)
Next, let's find the vertical component of the initial velocity (Vy). The vertical component of the final velocity (Vf) is given by:
Vy = Vf * sin(θ)
Plugging in the values, we get:
Vy = 8.70 m/s * sin(74.0°)
Now, we can find the magnitude of the initial velocity (V) using the Pythagorean theorem:
V = sqrt(Vx^2 + Vy^2)
Finally, to find the direction of the initial velocity, we can use trigonometric functions. Since both Vx and Vy are positive, the initial velocity is in the first quadrant.
Now, let's calculate the values:
Vx = 8.70 m/s * cos(74.0°)
Vy = 8.70 m/s * sin(74.0°)
V = sqrt(Vx^2 + Vy^2)
By plugging in the values and calculating, we can find the magnitude and direction of the initial velocity.
To find the initial velocity of the diver, we need to break down the given information.
Given:
- Height of the board (h) = 3 meters
- Speed upon contact with water (v) = 8.70 m/s
- Angle with respect to the horizontal surface of the water (θ) = 74.0°
Now, let's calculate the initial velocity (Vi) using the principle of conservation of mechanical energy.
Step 1: Determine the final velocity's vertical component.
When the diver contacts the water, her vertical velocity (Vy) is given by the equation:
Vy = v * sin(θ)
Vy = 8.70 m/s * sin(74.0°)
Vy = 8.70 m/s * 0.9613
Vy ≈ 8.36 m/s (rounded to 2 decimal places)
Step 2: Calculate the time taken to reach the water's surface.
To find the time of flight (t), we'll use the vertical component of velocity and the height of the board.
Using the equation of motion:
h = (1/2) * g * t^2
Where:
g = acceleration due to gravity (9.8 m/s^2)
Rearranging the equation:
t = sqrt((2h) / g)
t = sqrt((2 * 3) / 9.8)
t = sqrt(6 / 9.8)
t ≈ 0.78 s (rounded to 2 decimal places)
Step 3: Calculate the initial vertical velocity (Viy).
Using the equation of motion:
Vy = Viy - g * t
Plugging in the values:
8.36 m/s = Viy - 9.8 m/s^2 * 0.78 s
Viy = 8.36 m/s + (9.8 m/s^2 * 0.78 s)
Viy ≈ 15.45 m/s (rounded to 2 decimal places)
Step 4: Calculate the initial horizontal velocity (Vix).
The horizontal velocity remains constant throughout the motion.
Using the equation:
Vix = Vi * cos(θ)
Vix = Vi * cos(74.0°)
Vix = Vi * 0.2611
Step 5: Calculate the initial velocity (Vi).
To find the initial velocity (Vi), we can use the Pythagorean theorem:
Vi = sqrt(Vix^2 + Viy^2)
Vi = sqrt((Vi * 0.2611)^2 + (15.45 m/s)^2)
Solving this equation is a bit complex, involving iterative calculations. However, we can simplify the equation since the Vix value is relatively small compared to the Viy value. By neglecting the small term, we can approximate Vi as:
Vi ≈ Viy
Therefore: Vi ≈ 15.45 m/s
So, the initial velocity of the diver is approximately 15.45 m/s in the vertical direction, or in other words, straight upwards.