A diver leaves the end of a 4.0-m-high diving board and strikes the water 1.3 s later, 3.0 m beyond the end of the board. Considering the diver as a particle, determine:

(a) the horizontal component of the initial velocity

(b) the vertical component of the initial velocity

(c) the magnitude of the initial velocity

We will solve this problem using the kinematic equations.

(a) To find the horizontal component of the initial velocity, we will use the horizontal displacement formula:
x = x0 + vx * t

where x is the horizontal displacement, x0 is the initial horizontal position (which can be considered as 0), vx is the horizontal component of the initial velocity, and t is the time elapsed.

We are given x = 3.0 m and t = 1.3 s. Plugging these values into the equation, we get:

3.0 m = 0 + vx * 1.3 s

Now, we can solve for vx:

vx = 3.0 m / 1.3 s ≈ 2.31 m/s

Thus, the horizontal component of the initial velocity is approximately 2.31 m/s.

(b) To find the vertical component of the initial velocity, we will use the vertical displacement formula:
y = y0 + vy * t - 0.5 * g * t^2

where y is the vertical displacement, y0 is the initial vertical position (which can be considered as 0), vy is the vertical component of the initial velocity, g is the acceleration due to gravity (9.81 m/s^2), and t is the time elapsed.

We are given y = -4.0 m (negative because it is downward) and t = 1.3 s. Plugging these values into the equation, we get:

-4.0 m = 0 + vy * 1.3 s - 0.5 * 9.81 m/s^2 * (1.3 s)^2

Solving for vy, we get:

vy ≈ -0.87 m/s

Thus, the vertical component of the initial velocity is approximately -0.87 m/s (negative because it is downward).

(c) To find the magnitude of the initial velocity, we will use the Pythagorean theorem:
v = sqrt(vx^2 + vy^2)

Plugging in the values we found for vx and vy, we get:

v = sqrt((2.31 m/s)^2 + (-0.87 m/s)^2) ≈ 2.46 m/s

Thus, the magnitude of the initial velocity is approximately 2.46 m/s.

To solve this problem, we can use the kinematic equations of motion. We need to find the diver's initial velocity, time of flight, and horizontal velocity.

Let's consider the vertical motion of the diver first. We can use the equation:

y = y0 + v0yt + 0.5at^2

Where:
y = final vertical position (0 m as the diver strikes the water)
y0 = initial vertical position (4.0 m, the height of the diving board)
v0y = vertical component of initial velocity (unknown)
t = time of flight (1.3 s)

Substituting the values into the equation, we get:

0 = 4.0 m + v0y(1.3 s) - 0.5(9.8 m/s^2)(1.3 s)^2

Simplifying this equation will help us find the vertical component of the initial velocity, v0y.

Now let's consider the horizontal motion of the diver. We can use the equation:

x = x0 + v0xt

Where:
x = final horizontal position (3.0 m beyond the end of the diving board)
x0 = initial horizontal position (0 m)
v0x = horizontal component of initial velocity (unknown)
t = time of flight (1.3 s)

Substituting the values into the equation, we get:

3.0 m = 0 m + v0x(1.3 s)

Simplifying this equation will help us find the horizontal component of the initial velocity, v0x.

Finally, we can find the magnitude of the initial velocity, v0, using the Pythagorean theorem:

v0 = sqrt(v0x^2 + v0y^2)

Solving these equations will give us the answers to the problem.

To determine the answer, we need to use the equations of motion for an object in free fall. We can assume that the only force acting on the diver is gravity.

1. Find the initial vertical velocity:
The vertical distance fallen by the diver is equal to the vertical displacement from the top of the diving board to the water surface, which is 4.0 m.
Using the formula for vertical displacement in free fall:

d = (1/2) * g * t^2

where d is the vertical distance fallen, g is the acceleration due to gravity, and t is the time of fall.

Rearranging the equation for t:

t = sqrt(2d/g)

Plugging in the values:

t = sqrt((2 * 4.0 m) / (9.8 m/s^2))
t ≈ 0.90 s

Therefore, the time it takes for the diver to fall from the end of the diving board to the water surface is approximately 0.90 s.

2. Find the horizontal velocity:
The horizontal displacement of the diver is 3.0 m, and the time of fall is 1.3 s.
The horizontal velocity can be determined using the formula:

v = d / t

Plugging in the values:

v = 3.0 m / 1.3 s
v ≈ 2.31 m/s

Therefore, the horizontal velocity of the diver is approximately 2.31 m/s.

3. Find the initial velocity:
The initial velocity of the diver is the vector sum of the horizontal and vertical velocities.
Using the Pythagorean theorem:

v_initial = sqrt((v_horizontal)^2 + (v_vertical)^2)

Plugging in the values:

v_initial = sqrt((2.31 m/s)^2 + (0 m/s)^2)
v_initial ≈ 2.31 m/s

Therefore, the initial velocity of the diver is approximately 2.31 m/s.

4. Find the vertical velocity at impact:
Since the horizontal velocity remains constant throughout the motion, we only need to determine the final vertical velocity.
Using the formula for final vertical velocity in free fall:

v_vertical_final = v_vertical_initial + g * t

Plugging in the values:

v_vertical_final = 0 m/s + (9.8 m/s^2) * 1.3 s
v_vertical_final ≈ 12.74 m/s

Therefore, the vertical velocity of the diver just before hitting the water is approximately 12.74 m/s.