HOW DO I DO THIS PLZ HELP I DON'T NEED U TO SOLVE I JUST NEED TO KNOW HOW
Salmon often jump waterfalls to reach their
breeding grounds.
Starting 2.65 m from a waterfall 0.584 m
in height, at what minimum speed must a
salmon jumping at an angle of 25.2
◦
leave the
water to continue upstream? The acceleration
due to gravity is 9.81 m/s
2
.
Answer in units of m/s
To determine the minimum speed a salmon must have to jump the waterfall, we can use the principles of projectile motion. Here's how you can solve this problem:
1. Identify the given information:
- Initial distance from the waterfall (d): 2.65 m
- Height of the waterfall (h): 0.584 m
- Launch angle (θ): 25.2°
- Acceleration due to gravity (g): 9.81 m/s^2
2. Break down the problem into horizontal and vertical components:
- The horizontal component (x-direction) remains constant throughout the motion.
- The vertical component (y-direction) is influenced by gravity.
3. Find the time of flight:
- The time it takes for the salmon to reach the maximum height and come back down is given by the formula: t = (2 * v * sin(θ)) / g
Here, v is the initial velocity.
4. Determine the maximum height reached by the salmon:
- The formula for the maximum height (H) is: H = v^2 * sin^2(θ) / (2 * g)
5. Calculate the total distance covered by the salmon:
- The total distance (D) is the sum of the initial distance from the waterfall and the horizontal distance (x) traveled during flight.
D = d + x
x = v * cos(θ) * t
6. Apply kinematic equations to solve for the minimum speed:
- Use the formula for vertical motion to find v:
H = v^2 * sin^2(θ) / (2 * g)
- In our case, H is the height of the waterfall that the salmon needs to clear.
Substitute the given values for H and θ to get the equation:
0.584 m = v^2 * sin^2(25.2°) / (2 * 9.81 m/s^2)
- Rearrange the equation and solve for v:
v^2 = (2 * 9.81 m/s^2 * 0.584 m) / sin^2(25.2°)
v = square root of [(2 * 9.81 m/s^2 * 0.584 m) / sin^2(25.2°)]
7. Evaluate the equation to determine the minimum speed, rounding to the appropriate number of significant figures:
- Plug the values into a calculator:
v ≈ 2.53 m/s
So, the minimum speed the salmon must have to clear the waterfall is approximately 2.53 m/s.
As you know,
y(x) = tanθ x - g/(2 (v*cos^2θ)^2) x^2
For your angle, that gives
y(x) = .47x - 5.99/v^2 x^2
We want this parabola to have a vertex at (2.65,0.584) so since its max is at x=0.039v^2, we need
.039v^2 = 2.65
v = 8.22
Check:
y(x) = .47x - .0886 x^2
has vertex at (2.65,0.623) which is high enough to clear the waterfall.