As a stop light changes from red to green, a car starts to cross through the intersection. An instant before it begins to move, its velocity is zero. Must its acceleration at that time also be zero. Why or why not? Explain.

Understanding how velocity changes over time is fundamental in physics, particularly when examining the motion of objects. This concept is referred to as the rate of change of velocity, commonly known as acceleration. Acceleration is defined mathematically as the derivative of velocity with respect to time, or simply how quickly velocity changes.

It's important to note that acceleration is a vector quantity, which means it has both magnitude and direction. An object can accelerate by increasing its speed, by decreasing its speed (also known as deceleration), or by changing its direction while keeping the same speed. For example, when a car moves forward and picks up speed, it is accelerating in the direction of motion. Conversely, if the car slows down, it has a negative acceleration (or deceleration) in the same linear direction.

In our exercise scenario, the acceleration of the car is an unseen force that causes the velocity to change from zero to a higher value. The moment the traffic light turns green, the driver pushes the gas pedal, which causes the car to accelerate from a standstill. This acceleration is critical, as it is the driving force that changes the state of the car's motion.

Motion from rest is a concept where an object transitions from being stationary to moving. Understanding how objects start moving from a rest position is key to studying kinematics, a branch of mechanics dealing with the motion of objects without considering the forces that cause the motion.

An object at rest has a velocity of zero. As we apply a force to the object, it generates acceleration, leading to a change in velocity. This change continues as long as the force is applied, and the object's velocity increases from zero to a positive value, representing motion. In our example with the car, it is momentarily at rest when the stoplight is red. As soon as the green light appears and the driver applies a force on the gas pedal, the car's engine generates the force needed to overcome static friction and start the wheels rolling, leading to motion from rest.

It's crucial to understand that while the car's initial velocity is zero, the acceleration is not necessarily zero. The moment a force acts on the car, it begins to accelerate, indicating a rate of change of velocity that is no longer zero.

The relationship between velocity and acceleration is at the heart of Newton's laws of motion. Acceleration is dependent on changes in velocity; whenever an object's velocity changes, whether in magnitude or direction, it is experiencing acceleration.

This relationship can be expressed through the equation: \[ a = \frac{\Delta v}{\Delta t} \] where \( a \) is acceleration, \( \Delta v \) is the change in velocity, and \( \Delta t \) is the change in time. Acceleration can thus be seen as the 'engine' that drives changes in velocity.

In the exercise we're discussing, even though the car is at rest initially, meaning its velocity (\( v \) ) is zero, the car does not need to have zero acceleration at that point. The moment the light turns green and the car begins to move, its changing velocity indicates that there is a nonzero acceleration involved. In physical terms, the car must exert force to overcome inertia and this force reflects an acceleration, a change in velocity over a specific period. This showcases that an object at rest can experience a positive acceleration, which in turn, initiates its motion. This fundamental concept helps us to predict and quantify motion in real-life situations.