Question

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 2 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]
(b) Estimate the instantaneous velocity of the particle when t=1.

Answer 1

The average velocity is calculated by finding the displacement over specific intervals and dividing by the time, while the instantaneous velocity at t=1 can be estimated by taking the derivative or observing the average velocity trend as intervals approach t=1.

Step-by-step explanation:

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 2 sin(πt)+2 cos(πt), where t is measured in seconds.

Average Velocity Calculation:

The average velocity over a time interval is calculated as the displacement divided by the time interval. For intervals [1, 2], [1, 1.1], [1, 1.01], and [1, 1.001], we evaluate the displacement equation at the start and end of each interval, take the difference, and divide by the interval length.

Instantaneous Velocity Estimation:

To estimate the instantaneous velocity of the particle when t=1, we can take the derivative of the displacement function and evaluate it at t=1. Alternatively, noticing that as the interval gets shorter, the average velocity approaches the instantaneous velocity, we can observe the trend in the average velocities as the intervals get smaller and approach t=1.