Respuesta :
Answer:[tex]A=23.57^{\circ}[/tex]
Explanation:
Given
velocity=100 ft/s
height of landing zone=10 ft
Equation of [tex]x(t)=100 t\cos (A)[/tex]
[tex]y(t)=-16t^2+100t\sin (A)+10[/tex]
Maximum height=35 feet
at maximum height
[tex]\frac{\mathrm{d} Y}{\mathrm{d} t}=0[/tex]
[tex]\frac{\mathrm{d} x}{\mathrm{d} t}=-32t+100\sin A[/tex]
[tex]t=3.125\sin A[/tex]
At [tex]t=3.125\sin A[/tex]
[tex]Y(t)=35=-16\times (3.125\sin A)^2+100\times 3.125\sin A\times \sin A+10[/tex]
[tex]312.5\sin^2 A-156.25\sin^2 A=25[/tex]
[tex]sin^2 A=0.16[/tex]
[tex]A=23.57^{\circ}[/tex]
To have a maximum height of 35 feet above the landing zone, the angle of launch required is approximately 23.58°
The reason why the above angle is correct is given as follows:
The given parameters are;
The speed with which the cyclist is travelling, v = 100 ft./sec (constant speed)
Height of the end of the ramp above ground, h = 10 feet
Angle of inclination of the ramp = A₀
Parametric equation for the cyclist are;
x(t) = 100·t·cos(A)
y(t) = -16·t² + 100·t·sin(A) + 10
Required;
To find the required launch angle for a maximum height of 35 feet
Solution:
At maximum height, [tex]\dfrac{d(y(t))}{dt} =y'(t) = 0[/tex]
y'(t) = -32·t + 100·sin(A)
At maximum height, y'(t) = 0 = -32·t + 100·sin(A)
Therefore;
[tex]At \ maximum \ height \ t = \dfrac{100 \cdot sin(A)}{32}[/tex]
Making sin(A) the subject also gives;
[tex]At \ maximum \ height \ sin(A)= \dfrac{32\cdot t}{100} = \dfrac{8\cdot t}{25}[/tex]
From which we have;
[tex]Maximum \ height \ y(t) = -16 \cdot t^2 + 100\cdot t \cdot \dfrac{8 \cdot t}{25} + 10= 16 \cdot t^2 + 10[/tex]
When the height is 35 feet, we get;
[tex]Maximum \ height \ y(t) = 35 = 16 \cdot t^2 + 10[/tex]
Therefore;
[tex]For \ a \ maximum \ height \ of \ 35 \ feet, \ t =\sqrt{\dfrac{35 - 10}{16} } = \dfrac{5}{4}[/tex]
Which gives;
[tex]At \ maximum \ height \ t = \dfrac{5}{4} = \dfrac{100 \cdot sin(A)}{32}[/tex]
[tex]sin(A)= \dfrac{5 \times 32}{4 \times 100} = \dfrac{2}{5}[/tex]
[tex]\angle A =arcsin \left( \dfrac{2}{5} \right) \approx 23.58 ^{\circ}[/tex]
Therefore, if the cyclist wants to have a maximum height of 35 feet above the landing zone, then the required launch angle, A ≈ 23.58°
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