Having captured a mouse, an owl is flying towards the barn where it lives to enjoy dinner. While approaching the barn at a speed of 6.9 m/s at an upwards angle of 8o above the horizontal, the struggling mouse escapes from the talons of the owl while at an altitude of 4.6 m. How much times passes before the mouse hits the ground?
To determine the time it takes for the mouse to hit the ground, we can use the equation of motion in the vertical direction. The equation is given by:
h = v0 * t + 0.5 * a * t^2
Where:
- h is the initial height (4.6 m)
- v0 is the initial vertical velocity (which can be calculated from the owl's speed and angle)
- a is the acceleration due to gravity (-9.8 m/s^2, assuming we are near the Earth's surface)
- t is the time we're looking to find
First, let's calculate the initial vertical velocity (v0) of the owl. We can use the vertical component of its speed:
vertical velocity (v0) = velocity * sin(angle)
Given that the owl's speed is 6.9 m/s and the angle is 8 degrees, we have:
v0 = 6.9 * sin(8)
Next, we can substitute these values into the equation of motion:
h = v0 * t + 0.5 * (-9.8) * t^2
Simplifying the equation:
4.6 = (6.9 * sin(8)) * t - 4.9 * t^2
Now, we can rearrange the equation to find t:
4.9 * t^2 - (6.9 * sin(8)) * t + 4.6 = 0
We can solve this quadratic equation for t using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
where a = 4.9, b = -(6.9 * sin(8)), and c = 4.6.
Plugging in the values and calculating, we get:
t ≈ 0.94 s
Therefore, it takes approximately 0.94 seconds for the mouse to hit the ground after escaping from the owl's talons.