Final answer:
The midpoints of the car and motorcycle align when their positions are equal. To find the time and position, we equate the position equations and solve for time and position.
Step-by-step explanation:
The midpoints of the car and motorcycle align again when their positions are equal, meaning they have traveled the same distance from their starting point. To find the time and position when this happens, we can use the kinematic equations.
Let's assume that the car and motorcycle align at time t and position x. The initial position of both vehicles is x = 0, and they both travel in the positive direction of the x-axis.
Time:
Using the equation for position with constant acceleration:
- Car: xC = (1/2)ac·t^2
- Motorcycle: xM = (1/2)am·t^2
Setting the two equations equal to each other:
(1/2)ac·t^2 = (1/2)am·t^2
Since ac < am, we know that the motorcycle will reach the midpoint first. To determine the time when the midpoint is aligned, we set the position equation of the car equal to the position equation of the motorcycle:
(1/2)ac·t^2 = (1/2)am·t^2
Simplifying:
ac·t^2 = am·t^2
Solving for t:
t = sqrt(ac/am)
Position:
To find the position, we substitute the value of t back into the position equation:
xC = (1/2)ac·(sqrt(ac/am))^2
xM = (1/2)am·(sqrt(ac/am))^2
xM= xC