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 pi t + 3 cos pi t, where t is measured in seconds.
(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.
Can someone please help me with this question, the trig functions have me confused. Please and thanks again.
PUt in the times given, first the final time,then calculate position, then put in the initial time, and calculate position.
avg velocity=(finalposition-initialposition)/timeduration
but how do we treat the cos and sin...just input them like that into the calculator or use radians?
and also for (b) is the answer -2pi
i got -3 for all part a can anyone verify that answer please
To find the average velocity during each time period, we need to find the displacement of the particle at the endpoints of the interval and then divide it by the duration of the interval.
(a) Find the average velocity during each time period:
(i) [1, 2]
To find the displacement, we substitute the values of t into the equation of motion:
s(1) = 2sin(π * 1) + 3cos(π * 1)
= 2sin(π) + 3cos(π)
= 0 + (-3)
= -3
s(2) = 2sin(π * 2) + 3cos(π * 2)
= 2sin(2π) + 3cos(2π)
= 0 + 3
= 3
The displacement during the interval [1, 2] is s(2) - s(1) = 3 - (-3) = 6.
The duration of the interval is 2 - 1 = 1.
Average velocity = displacement / duration = 6 / 1 = 6 cm/s.
(ii) [1, 1.1]
s(1) = 2sin(π * 1) + 3cos(π * 1) = -3
s(1.1) = 2sin(π * 1.1) + 3cos(π * 1.1) ≈ -2.919
Displacement = s(1.1) - s(1) ≈ -2.919 - (-3) ≈ 0.081
Duration = 1.1 - 1 = 0.1
Average velocity = displacement / duration ≈ 0.081 / 0.1 ≈ 0.81 cm/s.
(iii) [1, 1.01]
s(1) = -3
s(1.01) ≈ -2.9997 (approximate value)
Displacement = s(1.01) - s(1) ≈ -2.9997 - (-3) ≈ 0.0003
Duration = 1.01 - 1 = 0.01
Average velocity = displacement / duration ≈ 0.0003 / 0.01 ≈ 0.03 cm/s.
(iv) [1, 1.001]
s(1) = -3
s(1.001) ≈ -2.99997 (approximate value)
Displacement = s(1.001) - s(1) ≈ -2.99997 - (-3) ≈ 0.00003
Duration = 1.001 - 1 = 0.001
Average velocity = displacement / duration ≈ 0.00003 / 0.001 ≈ 0.03 cm/s.
(b) To estimate the instantaneous velocity when t = 1, we need to find the derivative of the equation of motion with respect to t and evaluate it at t = 1.
Differentiating the equation of motion:
s = 2sin(πt) + 3cos(πt)
v = ds/dt = 2πcos(πt) - 3πsin(πt)
Evaluating v at t = 1:
v(1) = 2πcos(π * 1) - 3πsin(π * 1)
= 2πcos(π) - 3πsin(π)
= 2π(-1) - 3π(0)
= -2π
The instantaneous velocity of the particle when t = 1 is -2π cm/s.