(II) A car traveling at a velocity v can stop in a minimum distance d. What would be the car’s minimum stopping distance if it were traveling at a velocity of 2v?

Verified step by step guidance

1. Identify the relationship between stopping distance and velocity. The stopping distance, d, is proportional to the square of the velocity, v. This can be expressed as d \propto v^2.
2. Set up the proportionality equation for the initial condition where the stopping distance is d when the velocity is v. This can be written as d = k \times v^2, where k is the proportionality constant.
3. Substitute the new velocity, 2v, into the proportionality equation to find the new stopping distance. Replace v with 2v in the equation, resulting in d_{new} = k \times (2v)^2.
4. Simplify the equation by calculating (2v)^2, which gives 4v^2. Thus, the equation becomes d_{new} = k \times 4v^2.
5. Recognize that since k \times v^2 = d, the equation for the new stopping distance can be rewritten as d_{new} = 4 \times d. This shows that the stopping distance quadruples when the velocity is doubled.

Key Concepts

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as velocity, acceleration, and displacement. In this context, understanding how velocity affects stopping distance is crucial, as it relates to the equations of motion that describe how an object behaves when it is in motion.

Stopping Distance

Stopping distance is the total distance a vehicle travels from the moment the brakes are applied until it comes to a complete stop. It is influenced by factors such as initial speed, reaction time, and braking force. The relationship between speed and stopping distance is quadratic, meaning that if the speed doubles, the stopping distance increases by a factor of four, which is essential for solving the given problem.

Energy and Work

The concepts of energy and work are fundamental in understanding how a vehicle stops. The kinetic energy of the car, which is proportional to the square of its velocity, must be dissipated through work done by the brakes to bring the car to a stop. This relationship highlights that as the speed increases, the energy that needs to be converted into work also increases significantly, affecting the stopping distance.