Salmon often jump upstream through waterfalls to reach their breeding grounds. One salmon came across a waterfall 1.30 m in height, which she jumped in 1.3 s at an angle of 72° to continue upstream. What was the initial speed of her jump? (Assume the launch angle is measured above the horizontal.)

x = v(ox)•t =v(o)•cosα•t,

h =v(oy) •t - g•t²/2 = v(o)•sinα•t- g•t²/2,

x = v(o)•cosα•t,
h + g•t²/2 = v(o)•sinα•t,
(2h+ g•t²)/2•x = sinα/cosα = tanα,

x =(2h+ g•t²)/2•tanα =
=(2•1.3 +9.8•1.3²)/2•tan72º= 3.11 m.

v(o) = x/cosα•t =3.11/cos72º•1.3 =7.74 m/s.

To find the initial speed of the salmon's jump, we can use the equations of projectile motion. The vertical component of velocity can be determined using the equation:

Vf = Vi + at

Where:
Vf = final vertical velocity (0 m/s, since the salmon landed at the same height it started)
Vi = initial vertical velocity (unknown)
a = acceleration due to gravity (-9.8 m/s²)
t = time taken to complete the jump (1.3 s)

We know that the salmon jumped at an angle of 72° relative to the horizontal. We can find the vertical component of the initial velocity with the equation:

Vi = V * sin(angle)

Where:
V = initial velocity (unknown)
angle = launch angle (72°)

Since the salmon jumped upstream, its horizontal velocity remains constant throughout the jump. We can use the equation for horizontal motion to find the horizontal component of the initial velocity:

Vh = Vi * cos(angle)

Where:
Vh = horizontal component of initial velocity
Vi = initial vertical velocity (calculated earlier)
angle = launch angle (72°)

Finally, we can calculate the magnitude of the initial velocity (V) using the Pythagorean theorem:

V = sqrt(Vh² + Vi²)

Let's plug in the values and calculate:

Vi = V * sin(angle)
Vi = V * sin(72°)

Vh = Vi * cos(angle)
Vh = V * sin(72°) * cos(72°)

V = sqrt(Vh² + Vi²)
V = sqrt[(V * sin(72°) * cos(72°))² + (V * sin(72°))²]

Simplifying the equation:

V = V * sqrt[(sin(72°) * cos(72°))² + sin²(72°)]
1 = sqrt[(sin(72°) * cos(72°))² + sin²(72°)]

Now we can solve for V:

1 = (sin(72°) * cos(72°))² + sin²(72°)

Using trigonometric identities, we know that sin²(72°) = (1 - cos(144°)) / 2:

1 = (sin(72°) * cos(72°))² + (1 - cos(144°)) / 2

Now we can solve this equation to find the value of V.