A bullet is fired with an initial velocity of 100 m/s at an angle of 53° from the level ground. Determine the amount of time the bullet spends in the air.
(Use 10 m/s for gravity)
*10 m/s ^2
Vertical velocity:
Vy = V₀ sin θ = 100 ∙ sin 53° = 79.8636
Height:
h = Vy ∙ t - g t² / 2
When the bullet touches the ground the height will be zero.
Vy ∙ t - g t² / 2 = 0
Add g t² / 2 to both sides
Vy ∙ t = g t² / 2
Divide both sides by t
Vy = g t / 2
Multiply both sides by 2
2 Vy = g t
2 Vy / g = t
t = 2 Vy / g
t = 2 ∙ 79.8636 / 10 = 15.97272
t ≈ 16 sec
To determine the time the bullet spends in the air, we can break down the initial velocity into its horizontal and vertical components.
The horizontal component of the initial velocity is given by:
Vx = V * cos(θ),
where Vx is the horizontal component, V is the initial velocity (100 m/s), and θ is the angle of the initial velocity (53°).
Substituting the given values:
Vx = 100 m/s * cos(53°).
The vertical component of the initial velocity is given by:
Vy = V * sin(θ),
where Vy is the vertical component, V is the initial velocity (100 m/s), and θ is the angle of the initial velocity (53°).
Substituting the given values:
Vy = 100 m/s * sin(53°).
Now, we can find the time it takes for the bullet to reach the maximum height. At maximum height, the vertical velocity becomes zero.
Using the equation:
Vy = Voy - g * t,
where Vy is the vertical component of initial velocity, Voy is the vertical component of final velocity, g is the acceleration due to gravity (10 m/s²), and t is the time.
Substituting the values:
0 = Vy - g * t,
0 = 100 m/s * sin(53°) - 10 m/s² * t.
Now, solve for t:
10 m/s² * t = 100 m/s * sin(53°),
t = (100 m/s * sin(53°)) / (10 m/s²).
Simplifying the equation:
t = 10 * sin(53°) s.
To find the total time the bullet spends in the air, we double the time it takes to reach the maximum height. This is because the time to rise to the maximum height is equal to the time to fall back to the initial level ground.
Therefore:
Total time in the air = 2 * t,
Total time in the air = 2 * (10 * sin(53°)) s.
Calculating this expression will give you the amount of time the bullet spends in the air.