Salmon often jump waterfalls to reach their

breeding grounds.
Starting downstream, 2.95 m away from a
waterfall 0.54 m in height, at what minimum
speed must a salmon jumping at an angle of
29.7

leave the water to continue upstream?
The acceleration due to gravity is 9.81 m/s
2
.
Answer in units of m/s.

u = V cos 29.7 the whole time

x = u t
Vi = V sin 29.7 up velocity at start
v = Vi - 9.81 t
h = Vi t - (9.81/2) t^2

at top, v =0
so
t = Vi/9.81
h = .54 = Vi (Vi/9.81) - (9.81/2)( Vi^2/9.81^2)
.54 = (1/2)Vi^2/9,81
1.08*9.81 = Vi^2
etc
get Vi
get t
get u =2.95/t
get V = u^2+Vi^2

ah

V = sqrt (u^2 + Vi^2)

sorry damon but what is the variable u is in this example

u = horizontal speed the whole time (no horizontal force so no horizontal acceleration

starting from 2.95 m downstream, a line at an angle of 29.7º

... passes over the waterfall 1.68 m above the stream

this means the salmon's trajectory can drop (1.68 - .54) m during the jump

1.14 = 1/2 g t^2 ... 2.28 / g = t^2 ... t = .48 s

the horizontal component of the launch velocity is ... 2.95 m / .48s

the launch velocity is ... (horizontal component) / cos(29.7º)

I misread it. R Scott did it right.

To solve this problem, we can use the principles of projectile motion. Let's break down the problem into two parts: the horizontal motion and the vertical motion.

First, let's consider the horizontal motion. Since there is no horizontal acceleration, the horizontal speed of the salmon will remain constant throughout the jump. We can find the horizontal speed by using the formula:

Horizontal speed = Distance / Time

In this case, the distance is 2.95 m. We need to find the time it takes for the salmon to reach this distance. The time can be determined using the vertical motion of the salmon.

Now, let's consider the vertical motion. We can find the time it takes for a salmon to reach its maximum height using the formula:

Vertical distance = (Initial vertical velocity * Time) + (0.5 * Acceleration due to gravity * Time^2)

In this case, the initial vertical velocity is zero when the salmon leaves the water. The vertical distance is the height of the waterfall, which is 0.54 m. The acceleration due to gravity is 9.81 m/s^2.

We can solve the above equation to find the time it takes to reach the maximum height.

Once we have the time, we can use the concept of symmetry in projectile motion to determine the total time of flight. The time to reach the maximum height is equal to the time to fall from the maximum height back down to the initial height.

Finally, we can use the total time of flight to calculate the minimum speed the salmon must jump at to continue upstream. This can be found using the formula:

Vertical velocity = Acceleration due to gravity * Time

Now, we have all the information we need to solve the problem. Let's calculate the answer.

First, let's find the time it takes to reach the maximum height:

Vertical distance = (0 * Time) + (0.5 * 9.81 * Time^2)
0.54 = 4.905 * Time^2

Solving for Time gives us:
Time = sqrt(0.54 / 4.905) = 0.3487 s

Since symmetry tells us the total time of flight is twice the time it takes to reach the maximum height, the total time of flight is:
Total time = 2 * 0.3487 = 0.6974 s

Now, let's calculate the minimum speed the salmon must jump at:

Minimum speed = Distance / Time
Minimum speed = 2.95 / 0.6974 = 4.228 m/s

Therefore, the salmon must jump with a minimum speed of 4.228 m/s to continue upstream.