The displacement (in meters) of a particle moving in a straight line is given by the equation of motion
s = 5/t2,
where t is measured in seconds. Find the velocity of the particle at times
t = a, t = 1, t = 2, and t = 3.
(a) Find the average velocity during each time period.
(i) [1, 2]
cm/s
(ii) [1, 1.1]
cm/s
(iii) [1, 1.01]
cm/s
(iv) [1, 1.001]
cm/s
(b) Estimate the instantaneous velocity of the particle when t = 1.
cm/s
To find the velocity of the particle at different times, we need to differentiate the equation of motion with respect to time. Let's differentiate the equation s = 5/t^2 to find the velocity.
a) Average velocity during each time period:
(i) [1, 2]
To find the average velocity, we need to calculate the change in displacement over the given time interval. The average velocity is the ratio of the change in displacement to the change in time.
For the interval [1, 2]:
Change in displacement = s(2) - s(1)
= 5/(2^2) - 5/(1^2)
= 5/4 - 5/1
= 5/4 - 20/4
= -15/4
Change in time = 2 - 1 = 1
Average velocity = Change in displacement / Change in time
= (-15/4) / 1
= -15/4
= -3.75 m/s
Therefore, the average velocity during the time interval [1, 2] is -3.75 m/s.
(ii) [1, 1.1]
For the interval [1, 1.1]:
Change in displacement = s(1.1) - s(1)
= 5/(1.1^2) - 5/(1^2)
= 5/1.21 - 5/1
= 4.1322 - 5
= -0.8678
Change in time = 1.1 - 1 = 0.1
Average velocity = Change in displacement / Change in time
= (-0.8678) / 0.1
= -8.678 m/s
Therefore, the average velocity during the time interval [1, 1.1] is -8.678 m/s.
(iii) [1, 1.01]
For the interval [1, 1.01]:
Change in displacement = s(1.01) - s(1)
= 5/(1.01^2) - 5/(1^2)
= 5/1.0201 - 5/1
= 4.9505 - 5
= -0.0495
Change in time = 1.01 - 1 = 0.01
Average velocity = Change in displacement / Change in time
= (-0.0495) / 0.01
= -4.95 m/s
Therefore, the average velocity during the time interval [1, 1.01] is -4.95 m/s.
(iv) [1, 1.001]
For the interval [1, 1.001]:
Change in displacement = s(1.001) - s(1)
= 5/(1.001^2) - 5/(1^2)
= 5/1.002001 - 5/1
= 4.995005 - 5
= -0.004995
Change in time = 1.001 - 1 = 0.001
Average velocity = Change in displacement / Change in time
= (-0.004995) / 0.001
= -4.995 m/s
Therefore, the average velocity during the time interval [1, 1.001] is -4.995 m/s.
b) Instantaneous velocity when t = 1:
To estimate the instantaneous velocity at t = 1, we need to find the derivative of the equation of motion with respect to time and evaluate it at t = 1.
Velocity = ds/dt
Differentiating s = 5/t^2 with respect to t:
ds/dt = d(5/t^2)/dt
= (d5/dt)(1/t^2) + (5)(d(1/t^2)/dt)
= 0(1/t^2) + 5(-2/t^3)
= -10/t^3
Substituting t = 1 into the equation:
Velocity = -10/1^3
= -10 m/s
Therefore, the estimated instantaneous velocity of the particle when t = 1 is -10 m/s.