Mayan kings and many school sports teams are named for the puma, cougar, or mountain lion Felis concolor, the best jumper among animals. It can jump to a height of 13 ft when leaving the ground at an angle of 40.4°. With what speed, in SI units, does it leave the ground to make this leap? (m/s)
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To find the speed at which the puma leaves the ground to make this leap, we can use the principles of projectile motion. The vertical component of the puma's velocity will give us the information we need.
The vertical motion of a projectile can be described by the equation:
Vf^2 = Vi^2 + 2g * (Δy)
Where:
Vf = final velocity (unknown)
Vi = initial velocity (unknown)
g = acceleration due to gravity (9.8 m/s^2)
Δy = change in vertical position (height)
We are given that the puma jumps to a height of 13 ft, which is equivalent to approximately 3.9624 meters.
Using this information and rearranging the equation, we have:
Vf^2 = Vi^2 + 2 * 9.8 * 3.9624
The initial velocity (Vi) is the speed at which the puma leaves the ground.
To solve for Vi, we can substitute the given launch angle of 40.4° into the equation:
Vi = Vf / sin(θ)
Where:
θ = launch angle (40.4°)
Now we can substitute this into the previous equation:
Vf^2 = (Vf / sin(40.4°))^2 + 2 * 9.8 * 3.9624
Simplifying the equation, we get:
Vf^2 = Vf^2 / sin^2(40.4°) + 77.592
Multiplying through by sin^2(40.4°), we get:
sin^2(40.4°) * Vf^2 = Vf^2 + 77.592 * sin^2(40.4°)
Rearranging the equation gives us:
Vf^2 (1 - sin^2(40.4°)) = 77.592 * sin^2(40.4°)
Using the identity sin^2(θ) + cos^2(θ) = 1, we can simplify further:
Vf^2 * cos^2(40.4°) = 77.592 * sin^2(40.4°)
Finally, we can solve for Vf:
Vf = sqrt(77.592 * sin^2(40.4°) / cos^2(40.4°))
Evaluating this expression will give us the speed at which the puma leaves the ground.
To determine the speed at which the puma, cougar, or mountain lion leaves the ground to make a 13 ft leap at an angle of 40.4°, we can use the principles of projectile motion.
The velocity of an object can be broken down into its horizontal and vertical components. In this case, we are interested in the vertical component, which influences the height the animal can jump.
Given the height (h) = 13 ft and the launch angle (θ) = 40.4°, we can find the initial vertical velocity (Vy) using the following equation:
Vy = V * sin(θ)
Where Vy is the vertical component of the velocity, V is the initial velocity, and θ is the launch angle.
Since the height is given in feet, we need to convert it to meters. 1 foot is approximately 0.3048 meters. So, the height can be converted as:
h = 13 ft * 0.3048 m/ft = 3.9624 meters.
Now, substituting the values we have:
3.9624 m = V * sin(40.4°)
Rearranging the equation to solve for V:
V = 3.9624 m / sin(40.4°)
Using a calculator, we can find that sin(40.4°) is approximately 0.6428.
Substituting this value into the equation:
V = 3.9624 m / 0.6428
Using division, we find:
V ≈ 6.163 m/s
Therefore, the puma, cougar, or mountain lion must leave the ground with a speed of approximately 6.163 m/s to make a 13 ft leap at an angle of 40.4°.
The vertical component of Velocity, V sin 40.4, must provide enough kinetic energy to lift the animal by an amount
H = 13 ft (3.96 m) and create the potential energy required for that height.
(1/2) M V^2 sin^2 40.4 = M g H
V^2 = 2 g H/(sin^2 40.4)
Solve for V