A person on a hang glider is spiraling upward due to rapidly rising air. The acceleration of the glider is given by a(t)=-3cost i - 3sint j +2 k. The glider departed (t=0) from the point (3,0,0) with a velocity of v(t)=3 j. Find the glider's position as a function of time, t.
To find the position as a function of time, we need to integrate the acceleration to get the velocity and again to get the position. We will use initial conditions of s(0) = (3,0,0) and v(0) = 3j.
Integrating a(t), we get:
v(t) = ∫ a(t) dt = ∫(-3cos(t) i - 3sin(t) j + 2k) dt
v(t) = -3sin(t) i + 3cos(t) j + 2t k
Using the initial condition v(0) = 3j, we get:
0 = -3sin(0) i + 3cos(0) j + 2(0) k
0 = 3j
j = 0
Therefore, the velocity only has a component in the x and z directions:
v(t) = -3sin(t) i + 2t k
Integrating v(t), we get:
s(t) = ∫ v(t) dt = ∫(-3sin(t) i + 2t k) dt
s(t) = 3cos(t) i + t^2 k + C
Using the initial condition s(0) = (3,0,0), we get:
(3,0,0) = 3cos(0) i + 0k + C
C = (3,0,0)
Therefore, the position function is:
s(t) = 3cos(t) i + t^2 k + (3,0,0)
Final answer: s(t) = \<3cos(t) + 3, 0, t^2>
To find the position function of the glider as a function of time, we need to integrate the given acceleration function.
Given acceleration: a(t) = -3cos(t) i - 3sin(t) j + 2k
Integrating the acceleration function with respect to time will give us the velocity function:
v(t) = ∫ a(t) dt
v(t) = -3∫cos(t) dt i - 3∫sin(t) dt j + 2∫dt k
v(t) = -3sin(t) i + 3cos(t) j + 2t k + C1
Given initial velocity: v(0) = 3j
Setting t = 0 in the velocity function:
v(0) = -3sin(0) i + 3cos(0) j + 2(0) k + C1
3j = 0 i + 3 j + C1
Comparing the j-components, we get:
3 = 3 + C1
C1 = 0
Therefore, the velocity function is:
v(t) = -3sin(t) i + 3cos(t) j + 2t k
To find the position function, we need to integrate the velocity function with respect to time:
r(t) = ∫ v(t) dt
r(t) = ∫(-3sin(t) i + 3cos(t) j + 2t k) dt
r(t) = -3∫sin(t) dt i + 3∫cos(t) dt j + 2∫t dt k
r(t) = 3cos(t) i + 3sin(t) j + t^2 k + C2
Given initial position: r(0) = (3, 0, 0)
Setting t = 0 in the position function:
r(0) = 3cos(0) i + 3sin(0) j + (0)^2 k + C2
(3, 0, 0) = 3i + 0 j + 0 k + C2
Comparing the i-components, we get:
3 = 3 + C2
C2 = 0
Therefore, the position function is:
r(t) = 3cos(t) i + 3sin(t) j + t^2 k
To find the glider's position as a function of time, t, we need to integrate the acceleration function, a(t), twice with respect to time to get the position function.
Let's break down the given acceleration function a(t) into its components:
a(t) = -3cost i - 3sint j + 2 k
Integrating the x-component of the acceleration function with respect to time (t) will give the velocity in the x-direction, v_x(t):
v_x(t) = ∫(-3cost) dt
To integrate -3cost, we use the formula for the integral of cos(x):
∫cos(x) dx = sin(x) + C
Applying this to our integral, we get:
v_x(t) = [-3sin(t) + C1]
Given that the glider's velocity at t=0 is 3 j, we can determine the constant C1:
v_x(0) = -3sin(0) + C1 = 0
C1 = 0
Therefore, the x-component of the velocity is:
v_x(t) = -3sin(t)
To find the x-coordinate of the glider's position as a function of time, we integrate v_x(t) with respect to time (t):
x(t) = ∫v_x(t) dt
Integrating -3sin(t) will give us:
x(t) = ∫(-3sin(t)) dt = 3cos(t) + C2
Given that the glider departed from the point (3, 0, 0) at t=0, we can find the value of constant C2:
x(0) = 3cos(0) + C2 = 3 + C2 = 3
C2 = 0
Therefore, the x-coordinate of the glider's position as a function of time, t, is:
x(t) = 3cos(t)
Similarly, we can find the y-coordinate of the glider's position as a function of time, t, by integrating the y-component of the acceleration function, a(t):
v_y(t) = ∫(-3sint) dt = 3cos(t) + C3
Since the glider's velocity in the y-direction at t=0 is 3, we can determine the constant C3:
v_y(0) = 3cos(0) + C3 = 3
C3 = 3
Therefore, the y-coordinate of the glider's position as a function of time, t, is:
y(t) = 3cos(t) + 3
Lastly, the z-coordinate of the glider's position remains constant at z=0, as there is no acceleration component along the z-axis.
Therefore, the glider's position as a function of time, t, is given by:
r(t) = x(t) i + y(t) j + z(k)
= (3cos(t)) i + (3cos(t) + 3) j + 0 k