A stone is thrown upward from the top of a building at an angle of 30.0° to the horizontal and with an initial speed of 20.0 m/s, the height of the building is 45.0 m. How long is the stone in flight?

t = 4.218 s : stone flight time

Explanation:

The stone describes a parabolic path.

The parabolic movement results from the composition of a uniform rectilinear motion (horizontal) and a uniformly accelerated rectilinear motion of upward or downward motion (vertical).

The equation of uniform rectilinear motion (horizontal) for the x axis is:

x = xi + vx*t Equation (1)

Where:

x: horizontal position in meters (m)

xi: initial horizontal position in meters (m)

t : time (s)

vx: horizontal velocity in m/s

The equations of uniformly accelerated rectilinear motion of upward (vertical) for the y axis are:

y = y₀ + (v₀y) * t - (1/2) * g*t² Equation (2)

vfy = v₀y - gt Equation (3)

Where:

y: vertical position in meters (m)

y₀ : initial vertical position in meters (m)

t : time in seconds (s)

v₀y: initial vertical velocity in m/s

vfy: final vertical velocity in m/s

g: acceleration due to gravity in m/s²

Data

v₀ = 20.0 ° m/s, at an angle α₀=30.0° above the horizontal

y₀ = 45.0 m

g = 9.8 m/s²

Calculation of the time it takes for the stone to hit the ground

v₀y = v₀*sinα = (20 m/s) * sin (30°) = 10 m/s

We replace data in the equation (2)

y = y₀ + (v₀y) * t - (1/2) * gt²

0 = 45 + (10) * (t) - (1/2) * (9.8) (t) ²

(4.9) (t) ² - (10) (t) - 45 = 0

We solve the quadratic equation:

t₁ = 4.218 s

t₁ = - 2.177 s

Time cannot be negative therefore t₁ = 4.218 s is the time that the stone remains in the airt.

t = 4.218 s : stone flight time