Cliff divers at Acapulco jump into the sea from a cliff 37.1 m high. At the level of the sea, a rock sticks out a horizontal distance of13.39 m. The acceleration of gravity is 9.8 m/s^2. With what minimum horizontal velocity must the cliff divers leave the top of the cliff if they are to miss the rock?

h = Vo*t + 0.5g*t^2 = 37.1 m.

0 + 4.9t^2 = 37.1
t^2 = 7.57
t = 2.75 s.

d = V*t = 13.9 m.
2.75V = 13.9
V = 5.05 m/s.

Well, to avoid the rock, these cliff divers must have some mad skills in both physics and precision. Let's figure out the minimum horizontal velocity they need.

First, we need to find the time it takes for the divers to fall from a height of 37.1 m. We can use the formula:

d = 0.5 * g * t^2,

where d is the height (37.1 m) and g is the acceleration due to gravity (9.8 m/s^2).

Rearranging the formula to solve for time:

t = sqrt(2d / g),

we can now calculate t:

t = sqrt(2 * 37.1 m / 9.8 m/s^2) ≈ 2.18 s.

Now, we can find the minimum horizontal velocity using the formula:

v = d / t,

where d is the horizontal distance (13.39 m) and t is the time (2.18 s).

v = 13.39 m / 2.18 s ≈ 6.15 m/s.

So, these fearless cliff divers need a minimum horizontal velocity of around 6.15 m/s to miss that pesky rock. Just make sure they don't overshoot and land in the fish market!

To find the minimum horizontal velocity required for the cliff divers to miss the rock, we can use the principle of projectile motion.

First, let's consider the vertical motion of the cliff divers. We can use the equation for vertical displacement:

s = ut + (1/2)at^2

where:
s = vertical displacement (37.1 m, the height of the cliff)
u = initial vertical velocity (0 m/s, as the divers start from rest)
a = acceleration due to gravity (-9.8 m/s^2, as it acts in the opposite direction of motion)
t = time

Substituting the known values into the equation, we get:

37.1 = 0 x t + (1/2)(-9.8)t^2
37.1 = -4.9t^2

Next, let's consider the horizontal motion. The horizontal distance traveled by an object in projectile motion is given by:

d = vt

where:
d = horizontal distance (13.39 m)
v = horizontal velocity
t = time (which is the same as the time calculated in the vertical motion)

Rearranging the equation to solve for the horizontal velocity, we have:

v = d / t

Now, we can substitute the known values of d and t into the equation:

v = 13.39 / t

To miss the rock, the divers' horizontal velocity must be high enough so that they reach the horizontal distance of 13.39 m before hitting the surface of the water. This means that their vertical displacement and horizontal distance covered should have the same time of flight.

So, we can equate the time from the vertical motion equation (37.1 = -4.9t^2) to the time from the horizontal motion equation (v = 13.39 / t):

-4.9t^2 = 13.39 / t

Rearranging the equation, we get:

4.9t^3 = -13.39

Now, we can solve for t:

t^3 = -13.39 / 4.9
t^3 = -2.7342

Taking the cube root of both sides, we find:

t ≈ -1.377

Since time cannot be negative, this result is invalid. It means that a horizontal velocity cannot be found for the divers to miss the rock.

Therefore, there is no minimum horizontal velocity that the cliff divers can have to miss the rock if they jump from the given height and distance.

To solve this problem, we need to find the minimum horizontal velocity required for the cliff divers to miss the rock.

Let's break down the problem:

1. We are given the height of the cliff, which is 37.1 m.
2. The horizontal distance between the cliff and the rock is given as 13.39 m.
3. The acceleration due to gravity is 9.8 m/s^2.

Now, let's consider the motion of the cliff divers. The vertical motion can be analyzed independently from the horizontal motion. We need to find the horizontal velocity.

First, let's calculate the time it takes for the cliff divers to fall from the top of the cliff to the level of the sea using the vertical motion equation:

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

Here, h is the height of the cliff (37.1 m), g is the acceleration due to gravity (9.8 m/s^2), and t is the time of descent.

Rearranging the equation, we get:

t = √(2h / g)

Now, we can calculate the time it takes for the divers to reach the level of the sea:

t = √(2 * 37.1 / 9.8)
t ≈ 2.422 seconds

Next, since the horizontal distance between the cliff and the rock is 13.39 m, we can calculate the horizontal velocity required to cover this distance in 2.422 seconds:

v = d / t

Here, d is the distance (13.39 m) and t is the time (2.422 s).

v = 13.39 / 2.422
v ≈ 5.52 m/s

Therefore, the minimum horizontal velocity the cliff divers must have to miss the rock is approximately 5.52 m/s.