A Chinook (King) salmon (Genus Oncorynchus) can jump out of water with a speed of 5.95 m/s. (See Problem 4.9, page 111 for an investigation of how the fish can leave the water at a higher speed that it can swim underwater.) If the salmon is in a stream with water speed equal to 1.45 m/s, how high in the air can the fish jump if it leaves the water traveling vertically upwards relative to the Earth?
To determine the maximum height the salmon can jump, we can use the principle of conservation of energy. The initial kinetic energy of the salmon when it jumps out of the water will be equal to its final potential energy at the maximum height.
First, let's find the velocity of the salmon relative to the Earth when it jumps out of the water. The salmon's speed relative to the water is 5.95 m/s, and the water speed is 1.45 m/s. Therefore, the velocity of the salmon relative to the Earth is the difference between these two speeds:
Velocity relative to Earth = Speed of salmon relative to water - Speed of water
= 5.95 m/s - 1.45 m/s
= 4.5 m/s
Now, let's calculate the maximum height the salmon can reach. The initial kinetic energy is given by KE = (1/2) * mass * velocity^2, where mass is the mass of the salmon and velocity is the velocity relative to the Earth.
The final potential energy at its highest point is given by PE = mass * g * height, where g is the acceleration due to gravity (approximately 9.8 m/s^2) and height is the maximum height.
According to the conservation of energy, the initial kinetic energy should be equal to the final potential energy:
KE = PE
(1/2) * mass * velocity^2 = mass * g * height
We can cancel out the mass from both sides:
(1/2) * velocity^2 = g * height
Rearranging the equation, we can solve for the maximum height:
height = (1/2) * velocity^2 / g
Substituting the values, we have:
height = (1/2) * (4.5 m/s)^2 / 9.8 m/s^2
Calculating this expression, we find:
height ≈ 1.03 meters
Therefore, the salmon can jump approximately 1.03 meters high in the air relative to the Earth's surface.