\begin{tabular}{l} QUESTION 8 (Start on a new page.) \ The brakes of a car are being tested on a straight, level tarred road. At the far end \ of the road is a stretch of sof sand to stop the car, should it not stop in time. The \ length of the road up to the sand is 192 m . The car accolerates uniformly at \ $$ 9,8 \mathrm{~m} . \mathrm{s}^{2} $$ from rest at the start of the road. After travelling $$ 122,5 \mathrm{~m} $$, the brakes are \ applied and the car slows down uniformly, coming to rest 10 m from the sand. \ 8.1 Define acceleration. \ 8.2 Calculate the time taken for the car to travel the first $$ 122,5 \mathrm{~m} $$. \ 8.3 What is the velocity of the car at the $$ 122,5 \mathrm{~m} $$ mark? \ 8.4 Calculate the acceleration of the car during the time that the brakes are \ 8.5 applied. \ $$ \begin{array}{l} ext { The speedometer of a car registers 'instantaneous speed'. How is }\end{array} $$ \ \hline instantaneous speed measured? \end{tabular}

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\begin{tabular}{l} QUESTION 8 (Start on a new page.) \ The brakes of a car are being tested on a straight, level tarred road. At the far end \ of the road is a stretch of sof sand to stop the car, should it not stop in time. The \ length of the road up to the sand is 192 m . The car accolerates uniformly at \ $$ 9,8 \mathrm{~m} . \mathrm{s}^{2} $$ from rest at the start of the road. After travelling $$ 122,5 \mathrm{~m} $$, the brakes are \ applied and the car slows down uniformly, coming to rest 10 m from the sand. \ 8.1 Define acceleration. \ 8.2 Calculate the time taken for the car to travel the first $$ 122,5 \mathrm{~m} $$. \ 8.3 What is the velocity of the car at the $$ 122,5 \mathrm{~m} $$ mark? \ 8.4 Calculate the acceleration of the car during the time that the brakes are \ 8.5 applied. \ $$ \begin{array}{l}	ext { The speedometer of a car registers 'instantaneous speed'. How is }\end{array} $$ \ \hline instantaneous speed measured? \end{tabular}

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**8.1 Definition of Acceleration**

Acceleration is the rate of change of velocity with respect to time. It is defined mathematically as
$$
a = \frac{\Delta v}{\Delta t}.
$$

**8.2 Time to Travel the First $$122.5\,\text{m}$$**

The car starts from rest ($$u = 0$$) and accelerates uniformly at $$a = 9.8\,\text{m/s}^2$$. The displacement is given by
$$
s = ut + \frac{1}{2}at^2.
$$
Substituting in the values:
$$
122.5 = 0 \cdot t + \frac{1}{2}(9.8)t^2 = 4.9\,t^2.
$$
Solving for $$t$$:
$$
t^2 = \frac{122.5}{4.9} = 25 \quad \Longrightarrow \quad t = 5\,\text{s}.
$$

**8.3 Velocity at the $$122.5\,\text{m}$$ Mark**

Using the equation
$$
v = u + at,
$$
with $$u = 0\,\text{m/s}$$, $$a = 9.8\,\text{m/s}^2$$, and $$t = 5\,\text{s}$$, we get
$$
v = 0 + 9.8 \times 5 = 49\,\text{m/s}.
$$

**8.4 Acceleration During Braking**

After $$122.5\,\text{m}$$, the brakes are applied and the car comes to rest $$10\,\text{m}$$ from the sand. Given that the total length of the road up to the sand is $$192\,\text{m}$$, the braking distance is
$$
s = 192 - 10 - 122.5 = 182 - 122.5 = 59.5\,\text{m}.
$$
Let the initial velocity at the start of braking be $$u = 49\,\text{m/s}$$ and the final velocity $$v = 0\,\text{m/s}$$. Using the equation
$$
v^2 = u^2 + 2as,
$$
we plug in the values:
$$
0 = (49)^2 + 2a(59.5).
$$
This gives:
$$
2a(59.5) = -(49)^2.
$$
Calculating $$49^2 = 2401$$, then
$$
a = \frac{-2401}{2 \times 59.5} = \frac{-2401}{119} \approx -20.17\,\text{m/s}^2.
$$
The negative sign indicates that the acceleration is in the direction opposite to the motion (i.e., deceleration).

**8.5 How Instantaneous Speed is Measured**

Instantaneous speed is measured at a specific moment in time. It is defined as the limit of the average speed over an infinitely small time interval:
$$
v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t}.
$$
In practice, a car's speedometer determines instantaneous speed by measuring the rate at which the wheels rotate and converting this rotational speed to linear speed based on the wheel's circumference.
**8.1 Acceleration** Acceleration is how fast the speed of an object changes over time. **8.2 Time to Travel $$122.5\,\text{m}$$** The car takes **5 seconds** to travel the first $$122.5\,\text{m}$$. **8.3 Velocity at $$122.5\,\text{m}$$** At the $$122.5\,\text{m}$$ mark, the car is moving at **49 meters per second**. **8.4 Acceleration During Braking** When the brakes are applied, the car slows down at an acceleration of approximately **-20.17 meters per second squared** (negative indicates deceleration). **8.5 Measuring Instantaneous Speed** Instantaneous speed is measured by a car's speedometer, which calculates the speed at any given moment by measuring the rotation of the wheels and converting it to linear speed based on the wheel's size.

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