An adult dolphin weighs around 1460 N. How fast must he be moving as he leaves the water vertically in order to jump to a height of 2.90 m? Ignore air resistance.

PE at top=KE at launch

mgh=1/2 m v^2
1460*2.90=1/2 (1460/9.8)v^2

solve for velocity v.

Well, I must say, this dolphin needs some serious jumping skills! Let's crunch the numbers and see how high it needs to be on the speedometer.

To calculate the speed, we can use the conservation of energy. The initial kinetic energy of the dolphin will be converted into potential energy when it reaches the maximum height.

The equation for potential energy is:
PE = mgh

Where:
m = mass of the dolphin (we'll need that in kg)
g = acceleration due to gravity (9.8 m/s^2)
h = maximum height (2.90 m)
PE = Potential Energy

To start, let's convert the weight of the dolphin into mass using Newton's second law (F = mg):
Weight = mg

1460 N = m * 9.8 m/s^2

m = 1460 N / 9.8 m/s^2

m ≈ 149 kg

Now that we know the mass, we can calculate the speed it needs to reach. At the top of the jump, all of the initial kinetic energy will have been converted into potential energy, so we can equate them:

Initial Kinetic Energy = Potential Energy

(1/2)mv^2 = mgh

Simplifying:

(1/2)v^2 = gh

v^2 = 2gh

v = √(2gh)

Substituting the values we have:

v = √(2 * 9.8 m/s^2 * 2.90 m)

v ≈ √(57.04) m/s

v ≈ 7.55 m/s

So, the adult dolphin needs to be moving at approximately 7.55 meters per second to jump to a height of 2.90 meters. That's one speedy dolphin! Just be careful if you're ever out swimming with them - they might be practicing for the Olympics!

To determine the speed at which the dolphin must be moving in order to jump to a certain height, we can use the principle of conservation of energy.

The principle of conservation of energy states that the total energy of a system remains constant, assuming no external forces are acting on it. In this case, we can consider the total mechanical energy of the dolphin, which consists of its kinetic energy (KE) and potential energy (PE).

Initially, when the dolphin is in the water, it has only kinetic energy. As it jumps vertically out of the water, it converts its kinetic energy into potential energy at the highest point of the jump.

The equation for kinetic energy is:
KE = (1/2)mv²,
where m is the mass of the dolphin and v is its velocity or speed.

The equation for potential energy is:
PE = mgh,
where m is the mass of the dolphin, g is the acceleration due to gravity (9.8 m/s²), and h is the height it jumps.

Since the dolphin starts with only kinetic energy and ends with only potential energy, we can equate these two energies:

(1/2)mv² = mgh.

Simplifying the equation by canceling out the mass:
(1/2)v² = gh.

Rearranging the equation to solve for v:
v² = 2gh.

Now, we can substitute the given values into the equation:
v² = 2 * 9.8 m/s² * 2.90 m.

Calculating:
v² = 56.84 m²/s².

Taking the square root of both sides to solve for v:
v ≈ √(56.84) m/s.

Therefore, the dolphin must be moving at approximately 7.54 m/s vertically as it leaves the water in order to jump to a height of 2.90 m.

you suk bob