A 1100 kg automobile is at rest at a traffic signal. At the instant the light turns green, the automobile starts to move with a constant acceleration of 3.0 m/s2. At the same instant a 2800 kg truck, traveling at a constant speed of 8.0 m/s, overtakes and passes the automobile. (a) How far is the center of mass of the automobile–truck system from the traffic light at t= 3.0s? (b) What is the speed of the center of mass of the automobile-truck system then?

First write the location of the auto and of the truck as a function of time after the light turns green (t). Call these locations X1 and X2.

X1 = 1.5 t^2 (the auto)
X2 = 8.0 t (the truck)

The center of mass location is
X3 = (1100 X1 + 2800 X2)/3900.

The speed of the CM is dX3/dt
V3 = [1100*3t + 2800*8]/3900

Finish the calculation

To solve this problem, we need to analyze the motion of both the automobile and the truck separately and then consider the motion of the center of mass of the system.

(a) First, let's calculate the distance the truck traveled in 3.0 seconds. Since the truck is moving at a constant speed of 8.0 m/s, the distance it travels can be calculated using the equation:

Distance = Speed × Time

Distance = 8.0 m/s × 3.0 s

Distance = 24.0 m

So, at t = 3.0 s, the truck has traveled 24.0 meters.

Next, let's find the distance the automobile has traveled in 3.0 seconds. The automobile starts from rest and moves with a constant acceleration of 3.0 m/s^2. We can use the kinematic equation to find the displacement:

Displacement = Initial Velocity × Time + 0.5 × Acceleration × Time^2

Since the automobile starts from rest, the initial velocity is 0 m/s:

Displacement = 0.5 × 3.0 m/s^2 × (3.0 s)^2

Displacement = 13.5 m

So, at t = 3.0 s, the automobile has traveled 13.5 meters.

Now, let's find the distance between the center of mass of the automobile and the traffic light. Since the truck overtakes and passes the automobile, we can assume that the center of mass of the system is located at the truck.

Therefore, the distance between the center of mass and the traffic light is equal to the distance traveled by the truck minus the distance traveled by the automobile:

Distance = 24.0 m - 13.5 m

Distance = 10.5 m

So, the center of mass of the automobile-truck system is 10.5 meters away from the traffic light at t = 3.0 s.

(b) To find the speed of the center of mass of the system at t = 3.0 s, we need to consider the velocities of both the automobile and the truck separately.

The velocity of the truck remains constant at 8.0 m/s, so the speed of the center of mass due to the truck is also 8.0 m/s.

The automobile starts from rest and moves with a constant acceleration of 3.0 m/s^2 for 3.0 seconds. The final velocity at t = 3.0 s can be calculated using the equation:

Final Velocity = Initial Velocity + Acceleration × Time

Final Velocity = 0 m/s + 3.0 m/s^2 × 3.0 s

Final Velocity = 9.0 m/s

Therefore, the speed of the center of mass of the automobile-truck system at t = 3.0 s is 9.0 m/s.