A car starts from rest at a stop sign. It accelerates at 4.0 m/s2 for 3 seconds, coasts for 2 s, and then slows down at a rate of 3.0 m/s2 for the next stop sign. How far apart are the stop signs?

You need to divide the problem into three parts:

a) accelerated motion

what we know: initial velocity vi = 0, time t = 3s, acceleration a = 40m/s²;

the formula to find the space is: s = vi + 1/2·a·t²

all the units of measurement are fine,

therefore: s = 0 + 1/2·4·9 = 18m

b) constant motion

we know t = 2s and that the velocity here is equal to the final velocity (vf) of part a), which we need to calculate:

vf = vi + a·t which brings vf = 0 + 4·3 = 12m/s;

we can know calculate the total space with the formula s = vf·t = 12·2 = 24m

c) decelerated motion

we know the acceleration a = - 3m/s² (minus because the velocity is decresing) and the initial velocity which is equals to the velocity of part b), therefore vi = 12m/s

we need to calculate first the time taken to bring the car to stop, we can use the formula t = v/a = 12/3 = 4s

we can know calculate the space through the formula

s = vi·t + 1/2·a·t² = 12·4 + 1/2· (-3) · (4²) = 48 - 24 = 24m

We can now sum up the three spaces found to get the total distance between the stop signs:

s = sa + sb + sc = 18+24+24 = 66m