A dive bomber has a velocity of 290 m/s at an angle θ below the horizontal. When the altitude of the aircraft is 2.15 km, it releases a bomb, which subsequently hits a target on the ground. The magnitude of the displacement from the point of release of the bomb to the target is 3.09 km. Find the angle θ.
Y = Vy(T) + (1/2)gT^2
X = Vx(T)
where
Y = distance of ground from point of release =2.15 km =2150 m
Vy = vertical component of the initial velocity = 290*sin Θ
Θ = angle of release with respect to the horizontal
T = time for bomb to reach the target
g = acceleration due to gravity = 9.8 m/sec^2 (constant)
X = horizontal displacement of bomb from release point = 3.09 km = 3090 m
Vx = horizontal component of initial velocity = 290(cos Θ)
Substituting appropriate values,
2150 = 290(sin Θ)(T) + (1/2)(9.8)T^2
and simplifying,
2150 = 290(sin Θ)(T) + 4.9(T^2) -- call this Equation 1
For the horizontal component,
3090 = 290(cos Θ)T
Solving for "T"
T = 3090/(cos Θ)(290)
T = 10.6551/cos Θ
and substituting this in Equation 1,
2150 = 290(sin Θ)(10.65/cos Θ) + 4.9(10.65/cos Θ)^2
I can't seem to simplify this? Can you help me?
To simplify the equation, we can start by multiplying throughout by (cos Θ):
2150cos Θ = 290sin Θ * 10.6551 + 4.9 * (10.6551)^2
Next, we can use the trigonometric identity sin 2Θ = 2sin Θ * cos Θ to rearrange the equation:
2150cos Θ = 20.6551 * 290sin Θ * cos Θ + (4.9 * 10.6551^2) * (2sin^2 Θ - 1)
Expanding further:
2150cos Θ = 20.6551 * 290sin Θ * cos Θ + (4.9 * 10.6551^2) * 2sin^2 Θ - (4.9 * 10.6551^2)
Rearranging the terms:
2150cos Θ - 20.6551 * 290sin Θ * cos Θ - (4.9 * 10.6551^2) * 2sin^2 Θ + (4.9 * 10.6551^2) = 0
Now, we can factor out common terms:
(2150 - 20.6551 * 290cos Θ) * cos Θ - (4.9 * 10.6551^2) * 2sin^2 Θ + (4.9 * 10.6551^2) = 0
Simplifying further:
2150cos Θ - 5984.921cos Θ + 104853.624sin^2 Θ - 104853.624 = 0
Combining like terms:
-3834.921cos Θ + 104853.624sin^2 Θ - 104853.624 = 0
We can further simplify by dividing throughout by -104853.624:
0.03659cos Θ - sin^2 Θ + 1 ≈ 0
Now, we have a quadratic equation in terms of sin Θ and cos Θ. However, we can make this equation easier to work with by using the identity sin^2 Θ = 1 - cos^2 Θ:
0.03659cos Θ - (1 - cos^2 Θ) + 1 ≈ 0
0.03659cos Θ - 1 + cos^2 Θ + 1 ≈ 0
Simplifying further:
0.03659cos Θ + cos^2 Θ ≈ 0
Now, we can solve this equation using numerical methods or by graphing the equation and finding the values of cos Θ that give a solution.