a body moves along a straight line,its distance from the origin at any instant being given by the equation x= 8t -3t squared, where x is in centimeter and t is in seconds. find the average velocity of the body in the interval from t=0 to t=1 sec,and in the interval from t=0 to t=4 sec.
To find the average velocity of the body in a given time interval, we need to calculate the total displacement of the body during that time interval and divide it by the duration of the interval.
Let's start by finding the displacement of the body during the interval from t=0 to t=1 sec:
1. Substitute t=0 into the equation x = 8t - 3t^2:
x = 8(0) - 3(0)^2
x = 0
2. Substitute t=1 into the equation x = 8t - 3t^2:
x = 8(1) - 3(1)^2
x = 8 - 3
x = 5
The displacement during this interval is 5 centimeters.
Now, let's find the average velocity in the interval from t=0 to t=1 sec:
Average Velocity = Displacement / Time Interval
Average Velocity = 5 cm / (1 sec - 0 sec)
Average Velocity = 5 cm / 1 sec
Average Velocity = 5 cm/sec
Therefore, the average velocity of the body in the interval from t=0 to t=1 sec is 5 cm/sec.
Next, let's find the displacement of the body during the interval from t=0 to t=4 sec:
1. Substitute t=0 into the equation x = 8t - 3t^2:
x = 8(0) - 3(0)^2
x = 0
2. Substitute t=4 into the equation x = 8t - 3t^2:
x = 8(4) - 3(4)^2
x = 32 - 3(16)
x = 32 - 48
x = -16
The displacement during this interval is -16 centimeters.
Now, let's find the average velocity in the interval from t=0 to t=4 sec:
Average Velocity = Displacement / Time Interval
Average Velocity = -16 cm / (4 sec - 0 sec)
Average Velocity = -16 cm / 4 sec
Average Velocity = -4 cm/sec
Therefore, the average velocity of the body in the interval from t=0 to t=4 sec is -4 cm/sec.