At the local swimming pool, the diving board is elevated h = 7.5 m above the pool surface and overhangs the pool edge by L = 2 m. A diver runs horizontally along the diving board with a speed of v0 = 3.4 m/s and then falls into the pool. Neglect air resistance. Use a Cartesian coordinate system with its origin at the position of the diver just before falling. Let the direction that the diver falls be the negative y direction.
Calculate the time, tw in seconds, it takes the diver to move off the end of the diving board to the pool surface.
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To calculate the time it takes for the diver to move off the end of the diving board to the pool surface, we can use the equation of motion in the y-direction. Here are the steps to find the time (tw):
Step 1: Determine the initial vertical position (y0) of the diver. Since the origin is at the position of the diver just before falling, the initial vertical position (y0) is 0 m.
Step 2: Calculate the vertical displacement (Δy) of the diver from the diving board to the pool surface. Δy can be calculated using the equation: Δy = h, where h is the height of the diving board. In this case, h = 7.5 m.
Step 3: Use the equation of motion in the y-direction to find the time (tw) it takes for the diver to move off the end of the diving board to the pool surface. The equation is:
Δy = y0 + v0y × t - (1/2) × g × t^2
Where:
- Δy is the vertical displacement (h).
- y0 is the initial vertical position (0 m).
- v0y is the initial vertical velocity of the diver.
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
- t is the time it takes for the diver to move off the end of the diving board to the pool surface (tw).
Step 4: Substitute the known values into the equation and solve for tw:
7.5 = 0 + v0y × tw - (1/2) × 9.8 × tw^2
Step 5: Rearrange the equation to solve for tw:
(1/2) × 9.8 × tw^2 - v0y × tw + 7.5 = 0
This is a quadratic equation. You can use the quadratic formula to solve for tw:
tw = (-b ± √(b^2 - 4ac))/(2a)
Where:
- a = (1/2) × 9.8 = 4.9 (coefficient of tw^2)
- b = -v0y = -v0 × sin(θ) (coefficient of tw)
- c = 7.5 (constant term)
- √(b^2 - 4ac) represents the square root of the discriminant.
Step 6: Substitute the values of a, b, and c into the quadratic formula and calculate tw. Remember to take the positive root since we are considering the time it takes for the diver to move off the diving board to the pool surface:
tw = (-(-v0y) ± √((-v0y)^2 - 4 × 4.9 × 7.5))/(2 × 4.9)
Now, substitute the value of v0 into v0y:
tw = (v0 × sin(θ) ± √((v0 × sin(θ))^2 - 4 × 4.9 × 7.5))/(2 × 4.9)
Finally, substitute the values of v0, sin(θ), and evaluate the expression to find tw.
To calculate the time it takes for the diver to move off the end of the diving board to the pool surface, we can use the kinematic equations of motion.
First, let's break down the motion into horizontal and vertical components.
Horizontal motion:
The horizontal speed (vx) remains constant throughout the motion since there is no external force acting in that direction.
Vertical motion:
The initial vertical velocity (vy0) is zero, as the diver is starting from rest vertically.
The acceleration due to gravity (g) is acting downward.
Now, we can use the equation of motion in the vertical direction:
y = y0 + vy0t + (1/2)gt^2
Since the initial vertical velocity is zero, the equation simplifies to:
y = (1/2)gt^2
where y is the vertical displacement and t is the time.
The vertical displacement, y, is given by the height h of the diving board.
y = h
Substituting the values:
h = (1/2)gt^2
Now, we can solve for the time, t.
Multiply both sides by 2:
2h = gt^2
Divide both sides by g:
(2h) / g = t^2
Taking the square root of both sides:
sqrt((2h) / g) = t
Now, substitute the values:
t = sqrt((2 * 7.5) / 9.8)
Calculating this:
t ≈ 1.36 seconds
So, it takes approximately 1.36 seconds for the diver to move off the end of the diving board to the pool surface.