In attempting to jump up a waterfall, a salmon leaves the water 2.0 m from the base of the waterfall. With what minimum speed must it leave the water in order just to make it up a waterfall 1.6 m high?
To determine the minimum speed that the salmon must leave the water with in order to make it up the 1.6 m high waterfall, we can utilize the principle of conservation of energy.
The total mechanical energy of the salmon-waterfall system is conserved throughout the motion. At the bottom, the salmon only possesses kinetic energy, given by:
Kinetic Energy (KE) = (1/2) * mass * velocity^2 (Equation 1)
As the salmon reaches its highest point while ascending the waterfall, it will only possess gravitational potential energy. The gravitational potential energy at this point is given by:
Gravitational Potential Energy (PE) = mass * g * height (Equation 2)
where mass is the mass of the salmon, velocity is its velocity at the base, g is the acceleration due to gravity (approximately 9.81 m/s^2), and height is the height of the waterfall.
Since mechanical energy is conserved:
KE = PE
Setting Equations 1 and 2 equal to each other:
(1/2) * mass * velocity^2 = mass * g * height
Simplifying:
(1/2) * velocity^2 = g * height
Now, we can solve for the minimum speed (velocity) required for the salmon to just make it up the waterfall.
velocity = sqrt(2 * g * height)
Plugging in the given values:
velocity = sqrt(2 * 9.81 m/s^2 * 1.6 m)
Calculating:
velocity ≈ 6.29 m/s
Therefore, the minimum speed the salmon must leave the water with in order to just make it up the 1.6 m high waterfall is approximately 6.29 m/s.
4.5
L=vₒ²•sin2α/g= vₒ²•2sinα•cosα/g
h= vₒ²•sin²α/2g,
L/h= 2•vₒ²•2sinα•cosα•g/ vₒ²•sin²α• g=
=4/tanα
tanα = 4h/L = 4•1.6/2=3.2
α=74⁰
vₒ=sqrt{2gh/sin²α }= ...