6) CRITICAL THINKING: Kinematic equations are only valid for situations with constant acceleration. Using Newton’s second law, explain some examples of non-constant acceleration?
Newton's second law of motion relates the acceleration of an object to the net force acting on it and its mass, and it can be expressed as F(net) = m * a. In situations with non-constant acceleration, the net force acting on the object is not constant. Here are a few examples that demonstrate non-constant acceleration:
1) Projectile motion: When an object is launched into the air at an angle, it follows a curved trajectory called projectile motion. The acceleration of the object changes continuously as both the vertical and horizontal components of the velocity change. The force of gravity acts vertically downwards and causes a constant acceleration in the vertical direction.
2) Circular motion: In circular motion, an object moves along a curved path at a constant speed, but its velocity is constantly changing due to the change in direction. Since velocity is a vector quantity, a change in direction implies a change in velocity. Therefore, the object experiences a non-constant acceleration towards the center of the circle, known as centripetal acceleration. This acceleration is always directed perpendicular to the motion.
3) Car accelerating from rest: When a car starts moving from rest, it experiences an initial acceleration due to the force provided by the engine. Initially, the force is greater than the resistance forces such as friction and air resistance, resulting in a greater acceleration. As the car gains speed, the resistance forces increase, and the net force decreases, causing a decrease in acceleration until it reaches a constant velocity.
4) Skydiving with air resistance: A skydiver experiences a non-constant acceleration due to the opposing force of air resistance. At first, as the skydiver falls, the force of gravity is greater than the air resistance, resulting in an increasing acceleration. However, as the speed of the skydiver increases, the air resistance also increases and eventually becomes equal to the force of gravity, leading to a constant terminal velocity where the acceleration becomes zero.
These examples illustrate situations in which the net force acting on an object is not constant, leading to non-constant acceleration. Thus, Newton's second law is a more general and flexible framework for describing motion compared to the kinematic equations, which are valid only for situations with constant acceleration.
To explain examples of non-constant acceleration using Newton's second law, we have to understand the relationship between force, mass, and acceleration. Newton's second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, it can be written as F = ma, where F is the net force, m is the mass, and a is the acceleration.
In situations with non-constant acceleration, it means that the net force acting on an object is changing over time. Let's explore some examples:
1. Object falling through a fluid: When an object falls through a fluid like air or water, it experiences resistance due to the fluid. The amount of resistance depends on the object's velocity, surface area, and the density of the fluid. Initially, the force of gravity is greater than the resistance force, resulting in an acceleration. As the object accelerates, the resistance force increases until it becomes equal to the force of gravity. At this point, the net force becomes zero, and the object reaches a terminal velocity, where the acceleration is zero.
2. Rocket propulsion: When a rocket launches into space, it experiences non-constant acceleration due to the changing mass and the force exerted by the expelled propellant. Initially, the rocket has a high mass and experiences a large amount of thrust. As the rocket burns fuel, its mass decreases, resulting in a decrease in the force of thrust. Consequently, the acceleration of the rocket decreases as well.
3. Car accelerating with variable throttle: When you drive a car, the amount of force applied to the accelerator pedal determines the engine power and, consequently, the force or torque delivered to the wheels. If you press the accelerator smoothly and maintain a constant force, the acceleration will be relatively constant. However, if you vary the pressure on the pedal, the net force will change accordingly, resulting in an acceleration that is not constant.
In these examples, the net force acting on the objects changes due to factors like resistance forces, changing mass, or variable applied forces. Therefore, the acceleration experienced by these objects is not constant, and it cannot be described using the kinematic equations, which are specifically derived for situations with constant acceleration.