When salmon head upstream to spawn, they often must make their way up a waterfal

Question

When salmon head upstream to spawn, they often must make their way up a waterfall. If the water is not moving too fast, the salmon can swim right up through the falling water. If the water is falling with too great a speed, the salmon jump out of the water to get to a place in the waterfall where the water is not falling so fast. When humans build dams that interrupt the usual route followed by the salmon, artificial fish ladders must also be built to allow the salmon to get back uphill to the spawning area. These fish ladders consist of a series of small waterfalls with still pools of water between them (see the figure). Assume that the water is at rest in the pools at the top and bottom of one "rung" of the fish ladder, that water falls straight down from one pool to the next, and that salmon can swin at 4.8 m/s with respect to the water.

When salmon head upstream to spawn, they often must make their way up a waterfall. If the water is not moving too fast, the salmon can swim right up through the falling water. If the water is falling with too great a speed, the salmon jump out of the water to get to a place in the waterfall where the water is not falling so fast. When humans build dams that interrupt the usual route followed by the salmon, artificial fish ladders must also be built to allow the salmon to get back uphill to the spawning area. These fish ladders consist of a series of small waterfalls with still pools of water between them (see the figure). Assume that the water is at rest in the pools at the top and bottom of one "rung" of the fish ladder, that water falls straight down from one pool to the next, and that salmon can swin at 4.8 m/s with respect to the water. What is the maximum height of a waterfall up which the salmon can swim without having to jump? If a waterfall is 1.9 m high, how high must the salmon jump to get to water through which it can swim? Assume that they jump straight up. What initial speed must a salmon have to jump the height found in part (b)? For a 0.8 m high waterfall, how fast will the salmon be swimming with respect to the ground when it starts swimming up the waterfall? m/s

Explanation / Answer

part A:

let the height of the waterfall be h

the potential energy of the water at top is converted into kinetic energy when the water reaches bottom

0.5 * m * v^2 = m * g * h

v^2/2*g = h

this velocity of water with respect to ground (v)

velocity of fish w.r.t water = velocity of fish with respect to ground + velocity of water is with respect to ground

If velocity of fish w.r.t ground is less than the velocity of water is with respect to ground then fish can't swim and its has to jump.

therefore for maximum height of waterfall

velocity of water with respect to ground = velocity of fish with respect to ground

2* velocity of water with respect to ground = velocity of fish w.r.t water

velocity of water with respect to ground = 4.8/2 =2.4 m/s

v^2/2*g = h

h=0.288 meters ( assuming g = 10 m/s)

part B:

from previous part we inferred that the salmon can start swimming if the height difference is 0.288 m

therefore the salmon must jump upto 1.9 - 0.288=1.612 meters to start swimming again.

part C:

As found in part A that the salmon must atleast have a velocity w.r.t ground = 2.4m/s when there is height difference of 0.288 m between its position and the top point to reach to top of waterfall.

Therefore at 1.612m from the bottom of the waterfall the salmon must have a velocity of 2.4 m/s

The kinetic energy is at the bottom of the waterfall is converted into potential energy and kinetic energy when it reaches a height of 1.612 m

K.E @ bottom = P.E @ 1.612 m + K.E @ 1.612 m

K.E @ bottom = (m * g * 1.612) + (0.5 * m * 2.4 * 2.4)

0.5 * m * v^2 = m * 16.12 + m * 2.88

v = 6.164 m/s

part D:

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