Final answer:
To find the car's x-component acceleration (ax(t)), we derive it from the velocity function (vx(t)). If acceleration is constant, ax(t) remains steady; in simple harmonic motion, it follows a cosine function linked to force and displacement.
Step-by-step explanation:
To find the x-component of the acceleration of the car as a function of time ax(t), we first need to know the velocity function of the car in the x-direction, vx(t). Once we have this function, we can obtain ax(t) by taking the first derivative of vx(t) with respect to time. If the acceleration is constant, which is often the case in problems involving one-dimensional motion with constant acceleration, then ax(t) would simply be equal to that constant value.
For instance, in the situation where the car has a constant acceleration of 5.0 m/s², then that is the value of ax(t) regardless of time, indicating that the car is increasing its velocity in the positive x-direction at a rate of 5.0 m/s². However, if we're dealing with more complex motion such as simple harmonic motion, ax(t) could be a function depending on time, often involving sine or cosine functions if we're describing oscillations as a result of forces like springs or pendulums.
If we're dealing with simple harmonic motion, according to Newton's second law, a(t) = F/m = -kx/m, where k is the spring constant and x is the displacement. Hence ax(t) would be proportional to the displacement and in the opposite direction, often resulting in a cosine function if starting from maximum displacement.