Starting from rest, a car accelerates at a constant rate, reaching \(88 \mathrm{km} / \mathrm{h}\) in 12 s. Find (a) its acceleration and (b) how far it goes in this time.

When a car starts from rest and picks up speed uniformly over time, it is said to be experiencing constant acceleration. Acceleration is the rate at which an object's velocity changes over time. If the rate of change is steady, the acceleration is constant. This is a fundamental concept in physics, especially when analyzing motion in a straight line.

In the given exercise, the car's constant acceleration is determined using one of the equations of motion, specifically the first equation where initial velocity (\( u \)) is zero, and time (\( t \)) and final velocity (\( v \)) are known. This equation (\( v = u + at \)) is rearranged to solve for acceleration (\( a \)), yielding a value that represents how quickly the car's speed increases every second.

Displacement refers to the distance moved in a specific direction. It is a vector quantity, which means it has both magnitude and direction. To calculate the displacement of an object in motion with constant acceleration, the second equation of motion is used. This equation (\( s = ut + \frac{1}{2}at^2 \)) incorporates the object's initial velocity (\( u \)), acceleration (\( a \)), and time (\( t \)) to give the total displacement (\( s \)).

In our exercise, the car starts from rest, implying the initial velocity (\( u \)) is zero, thus simplifying the equation since the term (\( ut \)) becomes null. The remaining part of the equation, \frac{1}{2}at^2, is then used to calculate how far the car travels while accelerating over the given time period.

Unit conversion is essential in physics problems to ensure consistency and accuracy in calculations. The SI unit for speed or velocity is meters per second (\( m/s \)), whereas many practical situations, such as driving a car, are measured in kilometers per hour (\( km/h \)). To convert from \( km/h \) to \( m/s \), we multiply the speed by the conversion factor 0.278, because 1 \( km/h \) is equivalent to 0.278 \( m/s \).

In the exercise, converting the car's velocity from \( km/h \) to \( m/s \) before using it in the equations of motion ensures that all units are consistent with the SI system, allowing for accurate calculation of the car's acceleration and displacement.