# Physics problem - Salmon jumping waterfall

• FlowiwGhar
FlowiwGhar
Homework Statement
Salmon often jump waterfalls to reach their
breeding grounds.
Starting downstream, 2.73 m away from a
waterfall 0.614 m in height, at what minimum
speed must a salmon jumping at an angle of
26.7◦
leave the water to continue upstream?
The acceleration due to gravity is 9.81 m/s^2.
Relevant Equations
N/A
m * g * h + (1/2) * m * v² = m * g * y

Simplifying the equation:

g * h + (1/2) * v² = g * y

Substituting the values:

g * 0.614 + (1/2) * v² = g * 2.73 * sin(26.7°)

Now, let's solve for v:

(1/2) * v² = g * 2.73 * sin(26.7°) - g * 0.614

v² = 2 * (g * 2.73 * sin(26.7°) - g * 0.614)

v = √(2 * (g * 2.73 * sin(26.7°) - g * 0.614))

Substituting the value of g = 9.81 m/s² and performing the calculations:

v ≈ √(2 * (9.81 * 2.73 * sin(26.7°) - 9.81 * 0.614))

v ≈ √(2 * (53.803 - 6.018))

v ≈ √(2 * 47.785)

v ≈ √95.57

v ≈ 9.78 m/s

FlowiwGhar said:
Homework Statement: Salmon often jump waterfalls to reach their
breeding grounds.
Starting downstream, 2.73 m away from a
waterfall 0.614 m in height, at what minimum
speed must a salmon jumping at an angle of
26.7◦
leave the water to continue upstream?
The acceleration due to gravity is 9.81 m/s^2.
Relevant Equations: N/A

m * g * h + (1/2) * m * v² = m * g * y

Simplifying the equation:

g * h + (1/2) * v² = g * y

Substituting the values:

g * 0.614 + (1/2) * v² = g * 2.73 * sin(26.7°)

Now, let's solve for v:

(1/2) * v² = g * 2.73 * sin(26.7°) - g * 0.614

v² = 2 * (g * 2.73 * sin(26.7°) - g * 0.614)

v = √(2 * (g * 2.73 * sin(26.7°) - g * 0.614))

Substituting the value of g = 9.81 m/s² and performing the calculations:

v ≈ √(2 * (9.81 * 2.73 * sin(26.7°) - 9.81 * 0.614))

v ≈ √(2 * (53.803 - 6.018))

v ≈ √(2 * 47.785)

v ≈ √95.57

v ≈ 9.78 m/s
Have you posted this because you want someone to check it or because you have some reason to think your answer is wrong?

haruspex said:
Have you posted this because you want someone to check it or because you have some reason to think your answer is wrong?
The answer is incorrect, so I'd like someone to check it please

There is nothing to check. Your formulas and the values you plugged in looks like some random manipulation of symbols. For example., what is the meaning of the variable labeled "y" in your first equation? What is the meaning of the first equation and how is relevant to the problem?

Judging from your equations, y-h is the height the salmon would clear the dam by were it not for gravity. How that leads to your equation is mysterious. E.g. you add the KE to the PE needed to reach the top of the dam (mgh), but the PE is partly spent on that.

From your energy conservation equation it appears that ##y-h## is the height which the salmon reaches with zero speed. Is that what is happening here? As you know from projectile motion problems, the horizontal component of the velocity is constant. That means that the speed is never zero unless the fish jumps straight up which is not the case here.

Furthermore, it seems that you have misunderstood how to relate the given quantities to each other. In your equation you substitute ##y=2.73\sin(26.7^{\circ}).## That is incorrect and makes no sense. Draw a diagram showing ##y##, ##h##, the 2.73 m horizontal distance and the roughly parabolic trajectory of the fish to guide your thinking.

I would carry @haruspex 's suggestion one step further and recommend that you write an equation for the parabolic trajectory that relates the vertical displacement to the horizontal displacement, the initial speed and the angle of projection.

I note that since the angle of projection and the displacement of the salmon are all specified, there is no real minimization to be done.

Last edited:

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## How do you calculate the minimum speed a salmon must have to jump over a waterfall?

To calculate the minimum speed a salmon must have to jump over a waterfall, you can use the kinematic equations of motion. Assuming the salmon jumps vertically, the minimum speed (v) can be calculated using the equation v = sqrt(2gh), where g is the acceleration due to gravity (9.8 m/s²) and h is the height of the waterfall.

## What factors influence the ability of a salmon to jump over a waterfall?

The ability of a salmon to jump over a waterfall is influenced by several factors, including the height of the waterfall, the speed and angle of the jump, the strength and muscle power of the salmon, water flow conditions, and any obstacles or turbulence in the water.

## Why is the angle of the jump important for a salmon jumping a waterfall?

The angle of the jump is important because it affects the trajectory and the distance the salmon can cover. An optimal angle, usually around 45 degrees, maximizes the horizontal and vertical components of the jump, allowing the salmon to reach the necessary height while also covering the required distance to clear the waterfall.

## How does water flow affect a salmon's ability to jump a waterfall?

Water flow can significantly affect a salmon's ability to jump a waterfall. Strong currents and turbulence can make it more challenging for the salmon to gain the necessary speed and maintain a stable trajectory. Conversely, calmer waters can provide a more stable environment for the salmon to execute a successful jump.

## Can all salmon species jump over waterfalls, or is it specific to certain species?

Not all salmon species have the same jumping ability. Species like the Chinook and Coho salmon are known for their strong jumping capabilities and are more likely to successfully jump over waterfalls. Other species may not have the same strength or agility and may struggle with higher waterfalls.

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