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, we can use the principles of projectile motion. The vertical component of the initial velocity (v0y) can be determined using the formula:
v0y = vy * sin(θ)
where vy is the final vertical velocity (0 m/s, as the puma reaches the highest point of its jump) and θ is the angle at which it jumps (40.4°).
Next, we need to calculate the value of vy. In projectile motion, the vertical velocity decreases uniformly until it reaches its peak, where it becomes zero. At that point, the only force acting on the puma is gravity, which causes it to fall back down. The vertical motion can be modeled using the equation:
0 = vy - g * t
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time it takes for the puma to reach the highest point of the jump. This can be calculated using the formula:
t = v0y / g
We can substitute the value of t back into the previous equation to solve for vy:
0 = vy - g * (v0y / g)
0 = vy - v0y
Simplifying, we find vy = v0y.
Finally, we can calculate the initial vertical velocity, v0y:
v0y = vy * sin(θ)
v0y = 0 * sin(40.4°)
v0y = 0 m/s
The horizontal component of the initial velocity (v0x) remains constant throughout the jump. We can calculate its value using the formula:
v0x = vx * cos(θ)
Since the puma leaves the ground horizontally, the angle θ is not relevant to the horizontal component. Therefore, we have:
v0x = vx
To find vx, we can calculate the horizontal distance the puma travels during the jump. It is assumed that the time taken to reach the peak is equal to the time taken to return back to the ground. Thus, the total time of flight (2t) can be calculated using the formula:
2t = 2 * v0y / g
The horizontal distance travelled (d) can be calculated using the formula:
d = v0x * 2t
Substituting the values we have:
d = vx * 2 * (v0y / g)
To calculate vx, we need to determine the total horizontal distance the puma covers during the jump. We know that the maximum height (h) it reaches is 13 ft, or approximately 3.9624 meters. We can use the formula for projectile motion to calculate the time taken to reach this height:
h = v0y * t - (1/2) * g * t^2
Substituting the values we have:
3.9624 = 0 * t - (1/2) * 9.8 * t^2
This is a quadratic equation which can be solved for t:
4.9 * t^2 = 3.9624
t^2 = 0.807346939
Taking the square root of both sides, we find:
t ≈ 0.898 s
Substituting the value of t into the equation for d:
d = vx * 2 * (v0y / g)
d = vx * 2 * (0 / 9.8)
d = 0 m
Since d = 0, we can conclude that the puma does not cover any horizontal distance during its jump. Hence, the initial horizontal velocity (v0x) is also 0 m/s.
Therefore, the puma leaves the ground with an initial velocity of 0 m/s in both the horizontal and vertical directions.
To find the speed at which the puma leaves the ground, we can use the principles of projectile motion. We know the vertical distance (height) the puma jumps and the launch angle. We can assume there is no air resistance.
The formula to calculate the speed of an object in projectile motion is:
v = √((2 * g * h) / sin^2(θ))
where:
v = velocity (speed) of the puma
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height of the jump
θ = launch angle
Using the given values:
h = 13 ft = 3.9624 meters (converted to meters)
θ = 40.4° (given)
Now we can substitute these values into the formula and solve for v:
v = √((2 * 9.8 * 3.9624) / sin^2(40.4°))
In order to use the formula, we need to convert the angle from degrees to radians:
θ(rad) = θ(deg) * (π / 180)
θ(rad) = 40.4° * (π / 180) = 0.7057 radians (approximately)
Now let's calculate the speed:
v = √((2 * 9.8 * 3.9624) / sin^2(0.7057))
After performing the calculations, the speed at which the puma leaves the ground to make this leap is approximately 9.84 m/s (rounded to two decimal places).