Salmon often jump waterfalls to reach their
breeding grounds.
Starting downstream, 3.34 m away from a
waterfall 0.634 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
.
Answer in units of m/s.
Why did the salmon bring a ladder to the party? Because it wanted to reach its "breeding grounds" on the dance floor!
To tackle this fishy problem, we need to find the minimum speed the salmon must jump with to continue upstream. Since the salmon is starting downstream and needs to overcome the waterfall's height, we can use the conservation of energy principle.
The initial energy of the salmon is its potential energy (mgh), where m is the mass, g is the acceleration due to gravity, and h is the height of the waterfall. The final energy of the salmon is its kinetic energy, which is given by (1/2)mv^2, where v is the velocity of the salmon.
Setting the initial energy equal to the final energy, we can calculate the minimum speed required:
mgh = (1/2)mv^2
Cancelling out the mass:
gh = (1/2)v^2
Simplifying:
v^2 = 2gh
Taking the square root of both sides:
v = √(2gh)
Plugging in the given values:
v = √(2 * 9.81 m/s^2 * 0.634 m)
Calculating:
v ≈ 3.18 m/s
So, the minimum speed the salmon must leave the water with to continue upstream is approximately 3.18 m/s. Hopefully, it doesn't miscalculate the jump and end up doing a salmon flop instead!
To determine the minimum speed at which the salmon must jump to continue upstream, we can use the principle of conservation of energy. The potential energy lost as the salmon jumps up the waterfall is converted into kinetic energy. At the minimum speed required, all of the potential energy is converted into kinetic energy, neglecting any losses due to friction or air resistance.
First, let's calculate the potential energy of the salmon at the starting position, using the height of the waterfall and the acceleration due to gravity:
Potential Energy = mass * g * height
where mass is the mass of the salmon (which we can assume to be constant) and g is the acceleration due to gravity.
Next, let's calculate the kinetic energy required at the minimum speed:
Kinetic Energy = 0.5 * mass * velocity^2
where mass is the mass of the salmon (constant) and velocity is the minimum jumping speed.
Since energy is conserved, we can equate the potential energy to the kinetic energy and solve for the minimum speed:
mass * g * height = 0.5 * mass * velocity^2
Canceling out the mass term:
g * height = 0.5 * velocity^2
Now, let's plug in the given values:
g = 9.81 m/s^2 (acceleration due to gravity)
height = 0.634 m (height of the waterfall)
9.81 * 0.634 = 0.5 * velocity^2
Simplifying the equation:
0.634 * 9.81 = 0.5 * velocity^2
6.22 = 0.5 * velocity^2
Dividing both sides by 0.5:
12.44 = velocity^2
Taking the square root of both sides:
velocity = √12.44 = 3.52 m/s
So, at a minimum, the salmon must jump with a speed of 3.52 m/s to continue upstream.
Salmon often jump waterfalls to reach their
breeding grounds.
Starting downstream, 1.88 m away from a
waterfall 0.262 m in height, at what minimum
speed must a salmon jumping at an angle of
43.5
◦
leave the water to continue upstream?
The acceleration due to gravity is 9.81 m/s
2
.
d = Vo^2*sin(2A)/g = 3.34 m.
Vo^2*sin53.4/9.81 = 3.34,
Vo^2*0.082 = 3.34,
Vo = 6.4 m/s.