A car initially traveling at 30 m/s passes a road sign located 750 meters ahead in a time of 12 seconds. When the car reaches the road sign, the driver applies the brakes and brings the car to rest at a traffic light located 900 meters ahead of the road sign. Determine the cars total time of motion in seconds.
We can break the motion of the car into two parts: the time it takes for the car to reach the road sign, and the time it takes for the car to reach the traffic light.
For the first part, we know that the car is traveling at a constant speed of 30 m/s for 12 seconds. Therefore, the distance traveled is equal to speed multiplied by time: 30 m/s * 12 s = 360 meters. So the car reaches the road sign in 12 seconds.
For the second part, we know that the car is starting from rest and needs to come to a stop at the traffic light. We can use the equation of motion v^2 = u^2 + 2as, where v is the final velocity (0 m/s), u is the initial velocity (30 m/s), a is the acceleration, and s is the distance traveled.
Since we want to find the time taken to come to a stop, we rearrange the equation to get t = (v - u) / a. Since v = 0 m/s and u = 30 m/s, the equation becomes t = -30 m/s / a.
We also know that the distance traveled during this time is 900 meters - 750 meters = 150 meters.
Substituting these values into the equation, we have -30 m/s / a = 150 meters. Rearranging the equation to solve for a, we get a = -30 m/s / 150 meters = -0.2 m/s^2.
Now we can substitute this value of a into the equation t = -30 m/s / a to find the time taken to come to a stop: t = -30 m/s / -0.2 m/s^2 = 150 seconds.
Therefore, the total time of motion for the car is 12 seconds + 150 seconds = 162 seconds.