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 find the maximum height the Chinook salmon can jump, we need to consider the relative velocities of the fish and the water. We can use the concept of relative velocity to solve this problem.
The fish is jumping vertically upwards relative to the Earth, which means its initial vertical velocity is equal to the velocity with which it jumps out of the water. Let's denote this velocity as V_fish.
The velocity of the water, V_water, is given as 1.45 m/s. Since the fish is jumping out of the water, we need to subtract the velocity of the water from the velocity of the fish to find the net vertical velocity of the fish.
Net vertical velocity = V_fish - V_water
The maximum height the fish can reach is achieved when its vertical velocity becomes zero. At this point, the fish will start falling back down due to gravity.
To find the maximum height, we need to find the time taken by the fish to reach its highest point. We can use the equation of motion for vertical motion:
Vertical displacement = Initial vertical velocity * Time taken + (1/2) * Acceleration * Time taken^2
Since the vertical displacement is zero (maximum height - starting height is zero), we can solve for the time taken.
0 = V_fish * Time taken + (1/2) * (-9.8 m/s^2) * Time taken^2
Simplifying the equation, we get:
(-4.9 m/s^2) * Time taken^2 + V_fish * Time taken = 0
This equation is a quadratic equation. We can solve it using the quadratic formula:
Time taken = (-B ± sqrt(B^2 - 4AC)) / (2A)
Where A = (-4.9 m/s^2), B = V_fish, and C = 0.
After finding the time taken, we can substitute it back into the equation we derived to find the maximum height:
Max height = V_fish * Time taken + (1/2) * (-9.8 m/s^2) * Time taken^2
Now, let's substitute the given values and calculate the result:
V_fish = 5.95 m/s
V_water = 1.45 m/s
A = -4.9 m/s^2
B = V_fish = 5.95 m/s
C = 0
Plugging these values into the quadratic formula for Time taken:
Time taken = (-V_fish ± sqrt(V_fish^2 - 4AC)) / (2A)
Time taken = (-5.95 ± sqrt((5.95)^2 - 4 * (-4.9) * 0)) / (2 * (-4.9))
Time taken = (-5.95 ± sqrt(35.4025)) / (-9.8)
Calculating the positive root gives us:
Time taken ≈ 1.086 seconds (rounded to three decimal places)
Next, we can substitute this value of Time taken into the equation to find the maximum height:
Max height = V_fish * Time taken + (1/2) * (-9.8 m/s^2) * Time taken^2
Max height = 5.95 m/s * 1.086 s + (1/2) * (-9.8 m/s^2) * (1.086 s)^2
Max height ≈ 5.463 m (rounded to three decimal places)
Therefore, the Chinook salmon can jump to a height of approximately 5.463 meters above the water surface if it leaves the water traveling vertically upwards relative to the Earth.