The position of an object is given by the expression s=5−2sint . Which of the following is the object’s jerk at time t=2π3 ? (1 point) Responses −1 - 1 −3–√ - 3 0 0 3–√ 3 1
To find the object's jerk, we need to find the third derivative of the position function with respect to time.
Given the position function s = 5 - 2sin(t), let's find the derivatives step by step:
First derivative:
s' = d/dt(5 - 2sin(t))
= -2cos(t)
Second derivative:
s'' = d/dt(-2cos(t))
= 2sin(t)
Third derivative:
s''' = d/dt(2sin(t))
= 2cos(t)
Now, we need to evaluate the third derivative at t = 2π/3:
s'''(2π/3) = 2cos(2π/3)
= 2cos(120°)
= 2(-1/2)
= -1
Therefore, the object's jerk at time t = 2π/3 is -1.
To find the jerk of an object at a specific time, we need to differentiate the position function twice with respect to time.
Given the position function:
s = 5 - 2sin(t)
First, we need to find the first derivative of s with respect to t:
ds/dt = d/dt(5 - 2sin(t))
ds/dt = 0 - 2cos(t)
Next, we need to find the second derivative of s with respect to t:
d^2s/dt^2 = d/dt(-2cos(t))
d^2s/dt^2 = 2sin(t)
Now, we can substitute t = 2π/3 into the second derivative:
d^2s/dt^2 = 2sin(2π/3)
To simplify this, recall that sin(2π/3) = sin(π - 2π/3) = sin(π/3) = √3/2.
Therefore, the jerk of the object at t = 2π/3 is:
d^2s/dt^2 = 2 * (√3/2)
Simplifying further:
d^2s/dt^2 = √3
So, the jerk of the object at time t = 2π/3 is 3√3.
Therefore, the correct answer is 3√3.
To find the jerk at time t=2π/3, we will need to differentiate the given position expression twice with respect to time.
Given: s = 5 - 2sin(t)
First, let's find the velocity by differentiating the position expression with respect to time:
v = ds/dt = d/dt(5 - 2sin(t))
v = -2cos(t)
Next, let's find the acceleration by differentiating the velocity expression with respect to time:
a = dv/dt = d/dt(-2cos(t))
a = 2sin(t)
Lastly, let's find the jerk by differentiating the acceleration expression with respect to time:
j = da/dt = d/dt(2sin(t))
j = 2cos(t)
Now, substituting t = 2π/3 into the expression for jerk:
j(t = 2π/3) = 2cos(2π/3)
Now, cos(2π/3) = -1/2
j(t = 2π/3) = 2 * (-1/2)
j(t = 2π/3) = -1
Therefore, the jerk at time t=2π/3 is -1.
Hence, the correct response is -1.