Salmon, swimming up the Fraser river to their spawning grounds, leap over all sorts of obstacles. The unofficial salmon-altitude record is an amazing 3.58 m jump. Assuming the fish took off at 45.0°, what was its speed on emerging from the water? Ignore friction.
This problem is driving me CRAZY. I've tried this many times, but I have a feeling I'm making it a lot harder than it really is. Could someone tell me which equations I need to use?
Let V be the initial velocity leaving the water. The vertical component of the velocity, V sin 45, must be enough to let the salmon reach a height of h = 3.58 m
Maximum altitude h is reached when
V sin 45 = g T
The maximum height reached is then
h = (1/2) V sin 45 T
= (1/2) [V sin 45]^2/g = (1/4)V^2/g
V =sqrt (4 g h) = 2 sqrt (g h)
THANK YOU!
v = 11.8 m/s
That is faster than the fastest human can run horizontally (10 m/s)!
To solve this problem, you can use the principles of projectile motion. The equation you need to use is the range equation, which relates the horizontal distance traveled by an object to its initial velocity and launch angle.
Here are the steps to solve the problem:
1. Start by breaking down the initial velocity into its horizontal and vertical components. Since we are given the launch angle, we can use trigonometry to find these components. The horizontal component is given by: Vx = V * cos(theta), and the vertical component is given by: Vy = V * sin(theta), where V is the initial velocity and theta is the launch angle.
2. To find the time of flight, we can use the vertical motion equation: Vy = Voy - gt, where Voy is the initial vertical velocity (which is zero since the fish starts from the water's surface) and g is the acceleration due to gravity. Rearranging this equation gives us: t = Vy / g.
3. Now, we can use the time of flight to calculate the horizontal distance traveled by the fish. The horizontal distance is given by: Dx = Vx * t.
4. Since the fish emerged from the water at a height of 3.58 m, we can consider this as the maximum height (h) reached during the vertical motion. Using the kinematic equation for vertical motion, h = Voy*t - (1/2) * g * t^2, we can solve for t when h = 3.58 m.
5. Finally, using the value of t from step 4, we can substitute it into the equation from step 3 to find the horizontal distance traveled. Rearranging the equation, we have: V = Dx / t.
By following these steps, you can determine the speed of the salmon as it emerges from the water.