A Chinook (King) salmon (Genus Oncorynchus) can jump out of water with a speed of 6.75 m/s. If the salmon is in a stream with water speed equal to 1.65 m/s, how high in the air can the fish jump if it leaves the water traveling vertically upwards relative to the Earth?
The velocity of the salmon leaving the water in earth coordinates is the vector sum of 6.75 m/s (relative to water) and 1.65 m/s (relative to earth). The 6.75 m/s is a hypotenuse of a right triangle for which the upward velocity leg is
sqrt[(6.75)^2 - (1.65)^2] = 6.55 m/s
Set (1/2)M V^2 = M g H, with V = 6.55 m's, and solve for H.
The M's cancel.
To determine how high the salmon can jump, we need to consider the fish's initial velocity and the effects of gravity.
We know the fish's speed relative to the water, which is 6.75 m/s. However, the fish's motion relative to the Earth will be the vector sum of its speed and the stream's speed.
Since the fish is jumping vertically upwards relative to the Earth, its final velocity when it leaves the water will be zero, as its upward motion will gradually slow down until it stops at the peak of its jump.
Let's calculate the salmon's final velocity relative to the Earth:
Final velocity relative to the Earth = Fish speed relative to the water + Stream speed
Final velocity relative to the Earth = 6.75 m/s + 1.65 m/s
Final velocity relative to the Earth = 8.4 m/s
Now, we can use the concept of projectile motion to determine the maximum height the fish can reach. At the peak of the jump, its final velocity will be zero. We can use the equation of motion:
Final velocity^2 = Initial velocity^2 + 2 * acceleration * displacement
Since the final velocity is zero at the peak, the equation becomes:
Initial velocity^2 = -2 * acceleration * displacement
Here, acceleration is the acceleration due to gravity (approximately 9.8 m/s^2), and we want to find the displacement (height) the fish reaches.
Plugging in the values:
0 = -2 * 9.8 m/s^2 * displacement
Simplifying the equation:
Displacement = 0 / (-2 * 9.8 m/s^2)
Displacement = 0 m
From the calculation, we find that the salmon does not reach any height in the air. This means that even with the fish's jump and the stream's speed, its vertical motion does not lift it off the ground.
To determine how high the salmon can jump, we can use the concept of relative velocities.
The relative velocity of the fish with respect to the Earth is the difference between its velocity in the water and the velocity of the water itself.
Relative velocity = Velocity of the fish - Velocity of the water
Relative velocity = 6.75 m/s - 1.65 m/s
Relative velocity = 5.1 m/s
Now, we can use the equation of motion to find the maximum height the fish can reach. We need to consider the initial vertical velocity of the fish as 5.1 m/s.
The equation of motion for vertical motion is:
Final velocity^2 = Initial velocity^2 + 2 * acceleration * height
Since the fish reaches its maximum height, the final velocity is zero. The acceleration due to gravity is approximately 9.8 m/s^2.
0^2 = (5.1 m/s)^2 + 2 * (-9.8 m/s^2) * height
0 = 26.01 m^2/s^2 - 19.6 m/s^2 * height
19.6 m/s^2 * height = 26.01 m^2/s^2
height = 26.01 m^2/s^2 / 19.6 m/s^2
height ≈ 1.33 meters
Therefore, the salmon can jump approximately 1.33 meters high in the air relative to the Earth.