Salmon often jump waterfalls to reach their breeding grounds.

Question

Starting 3.16 m from a waterfall 0.257 m in height, at what minimum speed must a salmon jumping at an angle of 37.9◦ leave the water to continue upstream? The acceleration due to gravity is 9.81 m/s2 .
Answer in units of m/s

Answer 1

To solve this problem, we can break the salmon's motion into horizontal and vertical components. We can then use the kinematic equations to find the minimum speed required.

Horizontal motion:
Let vx be the horizontal component of the salmon's initial velocity.
vx = v * cos(37.9)
The horizontal distance the salmon must travel is 3.16 m. Since the acceleration due to gravity does not affect horizontal motion, the time it takes to travel this distance can be found with the equation:
x = vx * t
3.16 m = v * cos(37.9) * t

Vertical motion:
Similarly, let vy be the vertical component of the salmon's initial velocity.
vy = v * sin(37.9)
The vertical distance the salmon rises is 0.257 m, and the acceleration due to gravity is downward. The equation for the vertical displacement is:
y = vy * t - 0.5 * g * t^2
0.257 m = v * sin(37.9) * t - 0.5 * 9.81 m/s^2 * t^2

Now we have two equations with two unknowns, t and v. We can solve these equations to find the value of v.

Divide the first equation by the second equation to eliminate t:
(3.16 m) / (0.257 m) = (v * cos(37.9) * t) / (v * sin(37.9) * t - 0.5 * 9.81 m/s^2 * t^2)

Cancel out the t term and simplify the equation:
(3.16 m) / (0.257 m) = (cos(37.9)) / (sin(37.9) - (0.5 * 9.81 m/s^2 * t^2)/(v * sin(37.9) * t))

Now get a common denominator and combine the terms:

3.16*cos(37.9) = 0.257*sin(37.9) - 3.16*0.5*9.81*t^2/v*sin(37.9)

Now isolate the v term:
v = (3.16*0.5*9.81*t^2) / (0.257*cos(37.9) - 3.16*sin(37.9))

Finally, plug in the value of t from the horizontal equation into the vertical equation and solve for v:

t = 3.16/(v*cos(37.9))
v = (3.16*0.5*9.81*(3.16/(v*cos(37.9))^2)) / (0.257*cos(37.9)-3.16*sin(37.9))

Solving for v, we find that the minimum speed required for the salmon to jump the waterfall is approximately 4.33 m/s.

So the salmon must leave the water at a minimum speed of 4.33 m/s to continue upstream.