Question

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s=3cos(𝜋t)+5cos(𝜋t), where t is measured in seconds. (Round answer to 2 decimal places)

(a) Find the average velocity during each time period.

(i) [1, 2]: 10 cm/s

(ii) [1, 1.1]: __ cm/s

(iii) [1, 1.01] __ cm/s

(iv) [1, 1.001] __ cm/s

(b) Estimaate the instantaneous velocity of the particle when t=1. 

__ cm/s

Answer 1

To find the average velocity during each time period, we need to calculate the displacement over that time interval and then divide it by the time elapsed. The average velocity formula is:

Average velocity = (change in displacement) / (change in time)

Given equation of motion: s = 3cos(irt) + Scos(7rt)

(a) 

(i) [1, 2]:

Average velocity = (displacement at t = 2 - displacement at t = 1) / (2 - 1)

Calculate displacement at t = 1:

s(1) = 3cos(1r) + Scos(7r) = 3cos(r) + Scos(7r)

Calculate displacement at t = 2:

s(2) = 3cos(2r) + Scos(14r)

Average velocity = (s(2) - s(1)) / (2 - 1)

Average velocity = (3cos(2r) + Scos(14r) - (3cos(r) + Scos(7r))) / (2 - 1)

(ii) [1, 1.1]:

Average velocity = (displacement at t = 1.1 - displacement at t = 1) / (1.1 - 1)

Calculate displacement at t = 1.1:

s(1.1) = 3cos(1.1r) + Scos(7 * 1.1r)

Average velocity = (s(1.1) - s(1)) / (1.1 - 1)

Average velocity = (3cos(1.1r) + Scos(7 * 1.1r) - (3cos(r) + Scos(7r))) / (1.1 - 1)

(iii) [1, 1.01]:

Average velocity = (displacement at t = 1.01 - displacement at t = 1) / (1.01 - 1)

Calculate displacement at t = 1.01:

s(1.01) = 3cos(1.01r) + Scos(7 * 1.01r)

Average velocity = (s(1.01) - s(1)) / (1.01 - 1)

Average velocity = (3cos(1.01r) + Scos(7 * 1.01r) - (3cos(r) + Scos(7r))) / (1.01 - 1)

(iv) [1, 1.001]:

Average velocity = (displacement at t = 1.001 - displacement at t = 1) / (1.001 - 1)

Calculate displacement at t = 1.001:

s(1.001) = 3cos(1.001r) + Scos(7 * 1.001r)

Average velocity = (s(1.001) - s(1)) / (1.001 - 1)

Average velocity = (3cos(1.001r) + Scos(7 * 1.001r) - (3cos(r) + Scos(7r))) / (1.001 - 1)

(b) To estimate the instantaneous velocity of the particle when t = 1, we need to find the derivative of the displacement equation with respect to time (t) and then evaluate it at t = 1.

Velocity v = ds/dt = d/dt(3cos(irt) + Scos(7rt))

Evaluate velocity at t = 1:

v(1) = d/dt(3cos(ir) + Scos(7r))

Note: The value of Scos(7r) does not depend on time (t), so its derivative is zero.

Therefore, the instantaneous velocity at t = 1 is v(1) = d/dt(3cos(ir)) = -3rsin(r)

Now, substitute r = 1 (since t = 1) to get the instantaneous velocity at t = 1:

v(1) = -3 * 1 * sin(1) ≈ -2.68 cm/s (rounded to 2 decimal places)