KINEMATICS O26

1. Analyze the Starting Condition

Let us analyze the motion of two adjacent cars, Car 1 (leading) and Car 2 (following).
Given:

• Initial separation $x_{sep} = 10 \text{ m}$.
• Acceleration $a = 2 \text{ m/s}^2$.
• Car 2 starts when the distance to Car 1 becomes $35 \text{ m}$.
• Maximum speed $v_{max} = 72 \text{ km/h} = 20 \text{ m/s}$.

2. Calculate the Time Delay ($\Delta t$)

Let Car 1 start at $t=0$. Car 2 starts at $t = \Delta t$.
At $t = \Delta t$, Car 2 is still at its initial position. The separation has increased from $10 \text{ m}$ to $35 \text{ m}$. This means Car 1 has traveled a distance $d = 35 – 10 = 25 \text{ m}$.

Using the kinematic equation $s = ut + \frac{1}{2}at^2$ for Car 1:

$$ 25 = 0 + \frac{1}{2}(2)(\Delta t)^2 $$ $$ (\Delta t)^2 = 25 \implies \Delta t = 5 \text{ s} $$

So, Car 2 repeats the motion of Car 1 with a delay of 5 seconds.

3. Calculate Final Separation

When both cars are moving at $v_{max}$, the separation becomes constant. The total separation is the initial separation plus the extra distance gained by Car 1 while it was moving faster than Car 2.

Method 1: Relative Velocity Integration
We divide the motion into intervals based on the graph:

• $t = 0$ to $5$ s: Car 1 accelerates, Car 2 is at rest.
  Gap increase = Distance Car 1 travels = $25 \text{ m}$.
  Current Gap = $10 + 25 = 35 \text{ m}$.
• $t = 5$ to $10$ s: Car 1 accelerates ($10 \to 20 \text{ m/s}$), Car 2 accelerates ($0 \to 10 \text{ m/s}$).
  Relative velocity $v_{rel} = v_1 – v_2$. Since both have same $a$, $v_{rel}$ is constant.
  $v_{rel} = a \times \Delta t = 2 \times 5 = 10 \text{ m/s}$.
  Gap increase = $10 \text{ m/s} \times 5 \text{ s} = 50 \text{ m}$.
• $t = 10$ to $15$ s: Car 1 is at $v_{max} = 20 \text{ m/s}$, Car 2 accelerates ($10 \to 20 \text{ m/s}$).
  Relative velocity decreases linearly from $10 \text{ m/s}$ to $0$.
  Average $v_{rel} = 5 \text{ m/s}$.
  Gap increase = $5 \text{ m/s} \times 5 \text{ s} = 25 \text{ m}$.

Total Separation: $$ S_{final} = S_{initial} + \Delta S_1 + \Delta S_2 + \Delta S_3 $$ $$ S_{final} = 10 + 25 + 50 + 25 = 110 \text{ m} $$

ALITER: $$ S_{final} = S_{initial} + \text{GAP(AREA OF PARALLELOGRAM)} $$ $$ S_{final} = 10 + 5\times 20 = 110 \text{ m} $$