Let's break this problem down into two distinct time intervals and use the equations of motion for each.

Interval 1: $0 \leq t \leq T$

During this interval, the car accelerates with a constant acceleration $\alpha$ in the positive $x$-direction. The equations of motion are:

1. Velocity: The velocity at any time $t$ is given by: $v(t) = v_0 + \alpha t$ Since the car starts at rest at $t = 0$, $v_0 = 0$, so: $v(t) = \alpha t$

2. Position: The position of the car at any time $t$ is: $x(t) = x_0 + v_0 t + \frac{1}{2} \alpha t^2$ Since the car starts at rest at $x = 0$ (i.e., $x_0 = 0$ and $v_0 = 0$), the position becomes: $x(t) = \frac{1}{2} \alpha t^2$

So, for $0 \leq t \leq T$:

• Velocity: $v(t) = \alpha t$
• Position: $x(t) = \frac{1}{2} \alpha t^2$

Interval 2: $t > T$

At $t = T$, the car doubles its acceleration to $2\alpha$. The velocity at the moment $t = T$ is given by: $v(T) = \alpha T$ For times $t > T$, the new acceleration is $2\alpha$, so we now treat this as a new motion problem with the car starting at velocity $v(T)$ and acceleration $2\alpha$. The equations of motion for this new interval are:

1. Velocity (for $t > T$): $v(t) = v(T) + 2\alpha (t - T)$ Substituting $v(T) = \alpha T$, we get: $v(t) = \alpha T + 2\alpha (t - T) = \alpha T + 2\alpha t - 2\alpha T = 2\alpha t - \alpha T$

2. Position (for $t > T$): The position $x(t)$ can be found using: $x(t) = x(T) + v(T)(t - T) + \frac{1}{2} 2\alpha (t - T)^2$ The position at $t = T$ is: $x(T) = \frac{1}{2} \alpha T^2$ Substituting $v(T) = \alpha T$ and simplifying, we get: $x(t) = \frac{1}{2} \alpha T^2 + \alpha T (t - T) + \alpha (t - T)^2$ Expanding the terms: $x(t) = \frac{1}{2} \alpha T^2 + \alpha T t - \alpha T^2 + \alpha (t^2 - 2Tt + T^2)$ Simplifying: $x(t) = \alpha t^2 - \alpha T t + \frac{1}{2} \alpha T^2$

Summary of Results:

• For $0 \leq t \leq T$:

  • Velocity: $v(t) = \alpha t$
  • Position: $x(t) = \frac{1}{2} \alpha t^2$

• For $t > T$:

  • Velocity: $v(t) = 2\alpha t - \alpha T$
  • Position: $x(t) = \alpha t^2 - \alpha T t + \frac{1}{2} \alpha T^2$

Would you like further explanations or clarification on any part?