Relative Motion of Two Cars: Constant Speed vs Constant Acceleration

We compare the motion of two cars starting from the same point:

• Car A moves with constant velocity $v_A = 20\,\text{m/s}$.
• Car B starts from rest with constant acceleration $a_B = 4\,\text{m/s}^2$.

We use the standard kinematic equations (with $t$ in seconds, $x$ in meters):

1. Car A (constant speed)

   • Initial position: $x_{A0} = 0$
   • Initial velocity: $v_A = 20\,\text{m/s}$
   • Acceleration: $a_A = 0$

   Position as a function of time:

$$x_A(t) = x_{A0} + v_A t = 20 t$$

2. Car B (starting from rest, constant acceleration)

   • Initial position: $x_{B0} = 0$
   • Initial velocity: $v_{B0} = 0$
   • Acceleration: $a_B = 4\,\text{m/s}^2$

   Position:

$$x_B(t) = x_{B0} + v_{B0} t + \tfrac12 a_B t^2 = 2 t^2$$

3. When does car B catch car A?

We set the positions equal:

$$x_A(t) = x_B(t) \quad \Rightarrow \quad 20t = 2 t^2$$

Rearrange:

$$2t^2 - 20t = 0 \quad \Rightarrow \quad 2t(t - 10) = 0$$

So either $t = 0$ (the start) or $t = 10$ s. The non-trivial solution is:

$$t_{\text{meet}} = 10\,\text{s}$$

The meeting position:

$$x_A(10) = 20 \cdot 10 = 200\,\text{m}\\ x_B(10) = 2 \cdot 10^2 = 200\,\text{m}$$

Both cars are 200 m from the start at $t = 10$ s.

4. Interpretation

• Initially, car A pulls ahead with its constant speed.
• Car B starts slowly but speeds up because of its acceleration.
• After around 10 seconds, car B's increasing speed allows it to catch up with car A.

5. What this visualization shows

The canvas shows a 1D motion diagram along a horizontal road:

• A dark background for contrast.
• Car A (cyan) moves at constant speed.
• Car B (pink) starts at rest and accelerates.
• A vertical neon line marks the meeting point where their positions are equal.
• A time slider and play speed let you explore the motion before and after they meet.
• Optional toggles let you display the trajectory curves $x_A(t)$ and $x_B(t)$ in a mini graph area, so you can see where the curves intersect at $t=10$ s.

You can also adjust the speeds and acceleration in the widgets to see how the meeting time and distance change, turning this into a general tool for constant-speed vs constant-acceleration comparisons.