SHOW WORK PLEASE!!!!!
The displacement (in centimetres) of a particle moving back and forth along a straight line is given by the equation of motion s = 4 sin(πt) + 2 cos(πt),
where t is measured in seconds. (Round your answers to two decimal places.) (a) Find the average velocity during each time period.
(i) [1, 2]
(ii) [1, 1.1]
(iii) [1, 1.01]
(iv) [1, 1.001]
s = 4 sin(πt) + 2 cos(πt)
I will do the 2nd one, you do the others in the same way
when t = 1, s = 4 sin π + 2 cos π = 0 - 2 = -2
when t = 1.1, s = 4 sin (1.1π) + 2 cos (1.1π) = -3.138 (make sure your calculator is set to radians)
∆s/∆t = (-3.138 + 2)/(1.1-1) = appr -11.38
To find the average velocity during each time period, we need to calculate the change in displacement divided by the change in time.
(a) Average velocity from t = 1 to t = 2
To find the change in displacement, we need to evaluate the displacement equation at t = 2 and subtract the displacement at t = 1.
s(2) = 4sin(π(2)) + 2cos(π(2)) = 4sin(2π) + 2cos(2π) = 0 + 2 = 2
s(1) = 4sin(π(1)) + 2cos(π(1)) = 4sin(π) + 2cos(π) = 0 - 2 = -2
Change in displacement: Δs = s(2) - s(1) = 2 - (-2) = 4
Change in time: Δt = 2 - 1 = 1
Average velocity: v = Δs/Δt = 4/1 = 4 cm/s
(i) Average velocity from t = 1 to t = 1.1
s(1.1) = 4sin(π(1.1)) + 2cos(π(1.1)) ≈ -0.08
s(1) = -2
Change in displacement: Δs = s(1.1) - s(1) ≈ -0.08 - (-2) ≈ 1.92
Change in time: Δt = 1.1 - 1 = 0.1
Average velocity: v = Δs/Δt ≈ 1.92/0.1 ≈ 19.2 cm/s
(ii) Average velocity from t = 1 to t = 1.01
s(1.01) = 4sin(π(1.01)) + 2cos(π(1.01)) ≈ -1.15
s(1) = -2
Change in displacement: Δs = s(1.01) - s(1) ≈ -1.15 - (-2) ≈ 0.85
Change in time: Δt = 1.01 - 1 = 0.01
Average velocity: v = Δs/Δt ≈ 0.85/0.01 ≈ 85 cm/s
(iii) Average velocity from t = 1 to t = 1.001
s(1.001) = 4sin(π(1.001)) + 2cos(π(1.001)) ≈ -1.185
s(1) = -2
Change in displacement: Δs = s(1.001) - s(1) ≈ -1.185 - (-2) ≈ 0.815
Change in time: Δt = 1.001 - 1 = 0.001
Average velocity: v = Δs/Δt ≈ 0.815/0.001 ≈ 815 cm/s
So, the average velocities are:
(i) Approximately 4 cm/s
(ii) Approximately 19.2 cm/s
(iii) Approximately 85 cm/s
(iv) Approximately 815 cm/s