a bootlenose dolphin jumps out of the water. the path the dolphin travels can be modeled by h=-0.2d^2+2d, where h represents the height of the dolphin in feet and d represents the horizontal distance.
a) assume that all energy is conserved. what is the initial velocity with which the dolphin leaves the water? round your answer to the nearest hundredth. (Disregard water resistance; g=32f/s^2
b) Convert the initial velocity to mph. Round your answer to the nearest hundredth. (1 mile= 5,280 feet)
h=-.2d^2+2d
dh/dt=-.4d dd/dt+2dd/dt
so at max height, dh/dt=0
0=(-.4d+2)dd/dt
but dd/dt is the horizontalveloicy, it can be anything. d=5ft
So without knowing horizontal velocity, I don't know how you can solve for initial velocity.
To find the initial velocity with which the dolphin leaves the water, we need to understand that the height of the dolphin is related to the horizontal distance by the given equation:
h = -0.2d^2 + 2d
Since we're dealing with energy conservation, we know that the initial kinetic energy is equal to the final potential energy. We can express this using the formulas:
Initial kinetic energy = (1/2)mv^2
Final potential energy = mgh
Where:
m = mass of the dolphin (which we don't have)
v = velocity of the dolphin (what we're trying to find)
g = acceleration due to gravity (given as 32 ft/s^2)
h = height of the dolphin at the peak of its jump (maximum value of h) = ?
To find the height at the peak of the dolphin's jump, we need to calculate the vertex of the parabolic equation h = -0.2d^2 + 2d.
The vertex form of a parabolic equation is h = a(d - h)^2 + k, where (h, k) is the vertex of the parabola.
Comparing the given equation to the vertex form, we have:
a = -0.2
h = ? (we're trying to find this)
k = 0
The x-coordinate of the vertex, h, can be found using the formula:
h = -b / (2a), where b is the coefficient of the d term in the equation.
In our case, b = 2.
h = -2 / (2 * -0.2)
h = 2 / 0.4
h = 5
So, the height at the peak of the dolphin's jump is 5 feet.
Now, we can proceed to find the velocity at the peak of the dolphin's jump.
Using the conservation of energy:
(1/2)mv^2 = mgh
Dividing both sides of the equation by m:
(1/2)v^2 = gh
Substituting the known values:
(1/2)v^2 = 32 * 5
Simplifying:
v^2 = 160
Taking square roots:
v = √160
v ≈ 12.65
Therefore, the initial velocity with which the dolphin leaves the water is approximately 12.65 ft/s. (Answer to part a)
To convert this velocity to mph (miles per hour), we need to multiply it by the conversion factor. We know that 1 mile equals 5280 feet.
Converting 12.65 ft/s to mph:
12.65 ft/s * 3600 s/hr * 1 mile / 5280 ft ≈ 8.61 mph
So, the initial velocity of the dolphin leaving the water is approximately 8.61 mph. (Answer to part b)