A driver Speeds up a pulse of Car in 1.15 s from 80 km/h to 105 km/h during a displacement of 88m at a constant acceleration.

The problem states the initial speed ($u$) is $100 \text{ km/h}$, the final speed ($v$) is $80 \text{ km/h}$, and the displacement ($s$) is $88 \text{ km}$. Although the problem says "speeds up", the final speed is less than the initial speed, which means the car is actually decelerating.

Convert speeds from $\text{km/h}$ to $\text{m/s}$ and displacement from $\text{km}$ to $\text{m}$:

$$u = 100 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{100000}{3600} \text{ m/s} = \frac{250}{9} \text{ m/s}$$ $$v = 80 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{80000}{3600} \text{ m/s} = \frac{200}{9} \text{ m/s}$$ $$s = 88 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} = 88000 \text{ m}$$

To find the constant acceleration ($a$) given initial velocity ($u$), final velocity ($v$), and displacement ($s$), we use the following kinematic equation:

$$v^2 = u^2 + 2as$$

$$\left(\frac{200}{9} \text{ m/s}\right)^2 = \left(\frac{250}{9} \text{ m/s}\right)^2 + 2 \cdot a \cdot (88000 \text{ m})$$ $$\frac{40000}{81} \text{ m}^2/\text{s}^2 = \frac{62500}{81} \text{ m}^2/\text{s}^2 + 176000a \text{ m}$$

Subtract $\frac{62500}{81}$ from both sides of the equation:

$$176000a = \frac{40000}{81} - \frac{62500}{81}$$ $$176000a = \frac{40000 - 62500}{81}$$ $$176000a = \frac{-22500}{81}$$

Divide both sides by 176000:

$$a = \frac{-22500}{81 \times 176000}$$ $$a = \frac{-225}{81 \times 1760}$$

Simplify the fraction by dividing the numerator and denominator by 9:

$$a = \frac{-25}{9 \times 1760}$$ $$a = \frac{-25}{15840}$$

Simplify further by dividing the numerator and denominator by 5:

$$a = \frac{-5}{3168} \text{ m/s}^2$$

The negative sign indicates that the acceleration is in the opposite direction to the initial velocity, meaning it is a deceleration.

The acceleration of the car is $\mathbf{-\frac{5}{3168} \text{ m/s}^2}$.