A circus cat has been trained to leap off a 12-m-high platform and land on a pillow. The cat leaps off at
v0 = 3.6 m/s
and an angle
θ = 37°
and the question?
To solve this problem, we can break down the motion of the cat into horizontal and vertical components.
1. Calculate the vertical component of the initial velocity:
v0y = v0 * sin(θ)
= 3.6 m/s * sin(37°)
≈ 2.17 m/s
2. Calculate the time it takes for the cat to reach its maximum height:
The vertical motion of the cat can be modeled as free fall. The time it takes for the cat to reach its maximum height can be calculated using the equation:
vf = v0y + at
0 m/s = 2.17 m/s - 9.8 m/s^2 * t
t ≈ 0.22 s
3. Calculate the maximum height:
Using the equation for vertical displacement:
Δy = v0y * t + (1/2) * (-9.8 m/s^2) * t^2
= 2.17 m/s * 0.22 s + (1/2) * (-9.8 m/s^2) * (0.22 s)^2
≈ 0.47 m
4. Calculate the horizontal component of the initial velocity:
v0x = v0 * cos(θ)
= 3.6 m/s * cos(37°)
≈ 2.87 m/s
5. Calculate the horizontal distance traveled by the cat:
Using the equation for horizontal distance:
Δx = v0x * t
= 2.87 m/s * 0.22 s
≈ 0.63 m
Therefore, the cat will land on the pillow approximately 0.63 meters away from the base of the platform.
To analyze the motion of the circus cat, we can break down the initial velocity into its horizontal and vertical components.
The horizontal component (v₀x) remains constant throughout the cat's motion, while the vertical component (v₀y) changes due to the influence of gravity.
Using the given information, we can determine the initial velocity components:
v₀x = v₀ * cos(θ)
v₀y = v₀ * sin(θ)
Plugging in the given values:
v₀x = 3.6 m/s * cos(37°)
v₀x ≈ 2.88 m/s
v₀y = 3.6 m/s * sin(37°)
v₀y ≈ 2.16 m/s
Now, let's analyze the vertical motion of the cat. We can use the kinematic equation for vertical motion:
y = y₀ + v₀y * t - 0.5 * g * t²
Since the cat starts at a height of 12 m and lands on a pillow, the final height (y) will be 0. We want to find the time it takes for the cat to reach this height.
0 = 12 m + 2.16 m/s * t - 0.5 * 9.8 m/s² * t²
Rearranging the equation:
4.9 t² - 2.16 t - 12 = 0
Since this is a quadratic equation, we can solve it using the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
For this equation, a = 4.9, b = -2.16, and c = -12.
t = (-(-2.16) ± √((-2.16)² - 4 * 4.9 * -12)) / (2 * 4.9)
Simplifying and calculating:
t ≈ 1.226 s or t ≈ 1.755 s
Therefore, it would take approximately 1.226 seconds or 1.755 seconds for the cat to reach the pillow.