Question

If displacement ( in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s= 5 sin(πt)+5cos(πt), 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]

Answer 1

Explanation:

To find the average velocity during each time period, we can use the formula:

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

(i) [1,2]:

To find the change in displacement, we substitute the values of t into the equation of motion and subtract the displacement at t = 1 from the displacement at t = 2.

s(2) - s(1) = (5sin(π*2) + 5cos(π*2)) - (5sin(π*1) + 5cos(π*1))

Simplifying, we get:

s(2) - s(1) = 0 - (0 + 5)

s(2) - s(1) = -5

To find the change in time, we subtract the starting time from the ending time:

2 - 1 = 1

Now we can calculate the average velocity:

Average velocity = (-5) / 1 = -5 cm/s

(ii) [1,1.1]:

Using the same approach, we find the change in displacement:

s(1.1) - s(1) = (5sin(π*1.1) + 5cos(π*1.1)) - (5sin(π*1) + 5cos(π*1))

Simplifying, we get:

s(1.1) - s(1) = (0.984 - 4.99) - (0 + 5)

s(1.1) - s(1) = -4.006

The change in time is:

1.1 - 1 = 0.1

Calculating the average velocity:

Average velocity = (-4.006) / 0.1 = -40.06 cm/s

(iii) [1,1.01]:

Similarly, we find the change in displacement:

s(1.01) - s(1) = (5sin(π*1.01) + 5cos(π*1.01)) - (5sin(π*1) + 5cos(π*1))

Simplifying, we get:

s(1.01) - s(1) = (0.994 - 4.99) - (0 + 5)

s(1.01) - s(1) = -4.996

The change in time is:

1.01 - 1 = 0.01

Calculating the average velocity:

Average velocity = (-4.996) / 0.01 = -499.6 cm/s

(iv) [1,1.001]:

Again, we find the change in displacement:

s(1.001) - s(1) = (5sin(π*1.001) + 5cos(π*1.001)) - (5sin(π*1) + 5cos(π*1))

Simplifying, we get:

s(1.001) - s(1) = (0.999 - 4.99) - (0 + 5)

s(1.001) - s(1) = -4.991

The change in time is:

1.001 - 1 = 0.001

Calculating the average velocity:

Average velocity = (-4.991) / 0.001 = -4991 cm/s

In summary, the average velocities during each time period are:

(i) [1,2]: -5 cm/s

(ii) [1,1.1]: -40.06 cm/s

(iii) [1,1.01]: -499.6 cm/s

(iv) [1,1.001]: -4991 cm/s