Find the speed of a particle at time t = 8 whose position vector is vector r equals cosine times pi over 4 times t, 2 times sine of pi over 4 times t . (15 points)
A) 0
B) pi over 2
C) π
D) 3 times pi over 2
I guess you mean R = cos (pi t/4) i + 2 sin pi t/4 j
dR/dt = - (pi/4) sin ( pi t/4) i + (pi/2) cos (pi t/4)j
when t = 8
dR/dt = - pi/4 * 0 i + (pi/2) cos 2 pi j
cos 2 pi = -1
so dR/dt vector = 0 i - pi/2 j
|dR/dt| = speed = pi/2
B
To find the speed of a particle at time t=8, we need to calculate the magnitude of the velocity vector. The velocity vector is the derivative of the position vector with respect to time.
Given the position vector r = (cos(pi/4*t), 2*sin(pi/4*t)), we will differentiate each component of the vector separately.
1. Differentiating the first component:
The derivative of cos(pi/4*t) with respect to t can be found using the chain rule. The derivative of cos(x) is -sin(x), and the derivative of pi/4*t with respect to t is pi/4. Therefore, the derivative of cos(pi/4*t) is -sin(pi/4*t) * (pi/4).
2. Differentiating the second component:
Similarly, the derivative of 2*sin(pi/4*t) with respect to t can be found using the chain rule. The derivative of sin(x) is cos(x), and the derivative of pi/4*t with respect to t is pi/4. Therefore, the derivative of 2*sin(pi/4*t) is 2*cos(pi/4*t) * (pi/4).
Now we have the velocity vector:
v = (-sin(pi/4*t) * (pi/4), 2*cos(pi/4*t) * (pi/4))
The magnitude or speed of the particle is given by the formula:
speed = ||v|| = sqrt(v1^2 + v2^2)
Substituting the values of v1 and v2:
speed = sqrt((-sin(pi/4*t) * (pi/4))^2 + (2*cos(pi/4*t) * (pi/4))^2)
= sqrt((sin(pi/4*t))^2 + 4 * (cos(pi/4*t))^2 * (pi/4)^2)
= sqrt(sin^2(pi/4*t) + 4 * cos^2(pi/4*t) * (pi/4)^2)
Now, to find the speed at t=8, we substitute t=8 into the expression:
speed = sqrt(sin^2(pi/4*8) + 4 * cos^2(pi/4*8) * (pi/4)^2)
= sqrt(sin^2(2*pi) + 4 * cos^2(2*pi) * (pi/4)^2)
Since sin(2*pi) = 0 and cos(2*pi) = 1, we have:
speed = sqrt(0 + 4 * (1) * (pi/4)^2)
= sqrt(pi^2/4)
= pi/2
Therefore, the speed of the particle at t=8 is pi/2, which corresponds to option B) in the given choices.