a particle is projected from a point 0 with an initial speed of 30m/s to pass through a point which is 40m from 0 horizontally and 10m above 0. show that there are two angles of projection for which this is possible. if these angles are alpha and beta show that tan(alpha + beta) = -4
Whew.
The equation of a trajectory is
y = x tanθ - gx^2/(2*Vo*cos^2(θ))
y=0 ==> x=0
y=10 when x=40 ==>
10 = 40tanθ - 9.8x^2/(2*900*cos^2 θ)
10 = 40tanθ - 8.71sec^2 θ
10 = 40tanθ - 8.71 - 8.71tan^2 θ
8.71tan^2 θ - 40tanθ + 18.71 = 0
solving that for tanθ we get
tanθ = 0.5286 or 4.0638
Letting
a = arctan(.5286)
b = arctan(4.0638)
evaluating tan(a+b) = -4
To solve this problem, we can use the equations of motion to analyze the particle's motion in two dimensions: horizontal and vertical. Let's consider the horizontal and vertical components of the projectile separately.
1. Horizontal Motion:
The horizontal motion of the particle is uniform, with no acceleration. The equation we can use here is:
Horizontal distance = Initial horizontal velocity * Time
Given that the horizontal distance is 40m and the initial horizontal velocity is 30m/s, we can rearrange the equation to find the time taken for the horizontal motion:
Time = Horizontal distance / Initial horizontal velocity
Time = 40m / 30m/s
Time = 4/3s
2. Vertical Motion:
The vertical motion of the particle is under the influence of gravity. The equation we can use here is:
Vertical displacement = Initial vertical velocity * Time + (1/2) * acceleration * Time^2
Given that the vertical displacement is 10m, the initial vertical velocity is unknown, and the acceleration due to gravity is -9.8 m/s^2 (taking upward as positive), we can rearrange the equation to solve for the initial vertical velocity:
10m = Initial vertical velocity * (4/3s) + (1/2) * (-9.8m/s^2) * (4/3s)^2
10m = (4/3) * Initial vertical velocity - (4/3) * (1/2) * 9.8m/s^2 * (16/9)s^2
10m = (4/3) * Initial vertical velocity - (4/3) * 16.3m
10m = (4/3) * Initial vertical velocity - 21.74m
(4/3) * Initial vertical velocity = 21.74m + 10m
(4/3) * Initial vertical velocity = 31.74m
Initial vertical velocity = (31.74m) / (4/3)
Initial vertical velocity = 23.805m/s
Now that we have the initial vertical velocity, we can find the angles of projection.
3. Finding the Angles of Projection:
Let's assume the angles of projection are α and β. We can use the trigonometric relationships for projectile motion:
Initial horizontal velocity = Initial velocity * cos(angle of projection)
Initial vertical velocity = Initial velocity * sin(angle of projection)
Using these relationships, we can find the values of sin(α) and sin(β), and then the values of α and β.
Initial vertical velocity = 23.805m/s
Initial horizontal velocity = 30m/s
sin(α) = Initial vertical velocity / Initial velocity
sin(α) = 23.805m/s / 30m/s
sin(α) = 0.7935
α = arcsin(0.7935)
α ≈ 52.829°
sin(β) = Initial vertical velocity / Initial velocity
sin(β) = 23.805m/s / 30m/s
sin(β) = 0.7935
β = arcsin(0.7935)
β ≈ 52.829°
Therefore, we have found two angles of projection, α and β, both approximately equal to 52.829°.
4. Finding tan(α + β):
To find tan(α + β), we can use the trigonometric identity:
tan(α + β) = (tanα + tanβ) / (1 - tanα * tanβ)
tanα = tan52.829° ≈ 1.1082
tanβ = tan52.829° ≈ 1.1082
Using these values, we can calculate:
tan(α + β) = (1.1082 + 1.1082) / (1 - 1.1082 * 1.1082)
tan(α + β) = 2.2164 / (1 - 1.2283)
tan(α + β) = 2.2164 / (-0.2283)
tan(α + β) ≈ -4
Thus, we have shown that tan(α + β) is approximately equal to -4.
Therefore, there are two angles of projection, α and β, for which the particle can pass through the given point (40m horizontally and 10m vertically from the origin), and their sum gives tan(α + β) = -4.