The position of a particle moving in a straight line during a 2-second trip is given below.

s(t) = 2t 2 - 6t + 7 cm

(a) What is the average velocity for the entire trip? (Assume the 2 second interval starts at t = 0.)
1 cm/second

(b) Find a time at which the instantaneous velocity is equal to this average velocity.
2 seconds

go to sleep

disregard the 1 from cm/second and 2 from seconds thanks

To find the average velocity for the entire trip, divide the change in position by the change in time. In this case, the change in time is 2 seconds since the particle is moving for a 2-second trip. The change in position can be found by evaluating the position function at the endpoints of the interval.

(a) Average velocity = (s(2) - s(0)) / (2 - 0)

To find s(2), substitute t = 2 into the position function:
s(2) = 2(2)^2 - 6(2) + 7
= 8 - 12 + 7
= 3 cm

To find s(0), substitute t = 0 into the position function:
s(0) = 2(0)^2 - 6(0) + 7
= 0 - 0 + 7
= 7 cm

Average velocity = (3 - 7) / 2
= -4 / 2
= -2 cm/second

Therefore, the average velocity for the entire trip is -2 cm/second.

(b) To find a time at which the instantaneous velocity is equal to the average velocity, we need to find when the derivative of the position function is equal to -2 cm/second.

The derivative of the position function s(t) with respect to t gives the instantaneous velocity of the particle.

Taking the derivative of s(t):
s'(t) = 4t - 6

Setting s'(t) equal to -2:
4t - 6 = -2

Adding 6 to both sides:
4t = 4

Dividing both sides by 4:
t = 1

Therefore, at t = 1 second, the instantaneous velocity is equal to the average velocity of -2 cm/second.