Physics 9702 Study Notes: Non-Uniform Motion (Kinematics & Dynamics)
Hello future Physicists! Welcome to the exciting world of non-uniform motion. So far, you've mastered objects moving with constant acceleration (like a ball thrown straight up). But what happens when the forces fighting the motion—like air resistance—aren't constant? That's where non-uniform motion comes in! This chapter focuses on understanding these forces qualitatively, especially when objects reach their maximum speed: Terminal Velocity.
Don't worry if this sounds complicated. By the end of these notes, you'll be able to explain exactly why a tiny raindrop falls slower than a bowling ball, even though gravity is acting on both!
1. Understanding Resistive and Drag Forces
When an object moves, there are forces that always try to slow it down. These are called Resistive Forces.
1.1 Frictional and Viscous Forces
- Friction (Frictional Force): This is the force that resists motion between two solid surfaces in contact. Example: The friction between a car's tires and the road.
- Viscous/Drag Forces: These forces resist the motion of an object through a fluid (which means liquids or gases). We often call the resistive force exerted by air simply Air Resistance or Drag. Example: The force resisting a swimmer's motion through water, or a cyclist's motion through air.
For the purposes of this syllabus, we are focusing on the effect of viscous/drag forces (like air resistance) on moving objects.
Quick Review Box: Distinction
Friction: Solid vs. Solid (usually constant, or calculated using coefficients).
Drag/Viscous Force: Object vs. Fluid (depends entirely on speed).
1.2 The Simple Model: Drag Force Increases with Speed
The key qualitative insight you must master is how drag force changes with the object's velocity.
The Principle: A simple model shows that the magnitude of the drag force exerted by the fluid (air or water) increases as the speed of the object increases.
Analogy: Sticking your hand out the window
Imagine you put your hand out of a car window:
- If the car is moving slowly (low speed), you feel a gentle push (low drag).
- If the car speeds up (high speed), you feel a strong, uncomfortable push (high drag).
This demonstrates the qualitative understanding required: The faster you go, the harder the fluid pushes back.
Key Takeaway: Resistive forces (drag) are velocity-dependent. They oppose motion and get stronger as the speed increases.
2. Motion in a Uniform Gravitational Field with Air Resistance
Now, let's apply this understanding to a common situation: an object falling under gravity (like a skydiver or a falling baseball).
2.1 Forces Acting on a Falling Object
When an object falls towards the Earth, two primary forces are acting on it:
- Weight (\(W\)): The force of gravity pulling the object down. This force is constant and given by \(W = mg\), where \(m\) is the mass and \(g\) is the acceleration of free fall (approximately \(9.81 \, \text{m\,s}^{-2}\)).
- Air Resistance/Drag Force (\(F_{drag}\)): The viscous force opposing the motion (acting upwards). This force increases with speed.
The motion (acceleration) of the object is determined by the Resultant Force (\(F_{net}\)), following Newton's Second Law: \(F_{net} = ma\). The net force is \(W - F_{drag}\).
2.2 Explaining the Motion Qualitatively (The Skydiver Model)
Let's track the skydiver's journey from jumping out of the plane:
-
At the Start (Speed \(v\) = 0):
The skydiver has just jumped. Speed is zero, so drag force \(F_{drag}\) is zero.
Net Force: \(F_{net} = W - 0 = W\). This is maximum net force.
Acceleration: \(a\) is maximum (equal to \(g\)). The skydiver accelerates rapidly. -
During the Fall (Speed \(v\) is increasing):
As the skydiver speeds up, \(F_{drag}\) starts to increase (because drag depends on speed).
Net Force: Since \(F_{drag}\) is increasing, the resultant force \(F_{net} = W - F_{drag}\) is decreasing.
Acceleration: Since \(F_{net}\) is decreasing, the acceleration \(a\) is also decreasing. (The skydiver is still speeding up, just less rapidly). -
Reaching Terminal Velocity (Constant Speed):
The speed continues to increase until \(F_{drag}\) becomes equal in magnitude to the Weight \(W\).
Net Force: \(F_{net} = W - F_{drag} = 0\). The forces are balanced.
Acceleration: Since \(F_{net} = 0\), the acceleration \(a\) must also be zero.
Velocity: The object stops accelerating and continues falling at a constant, maximum speed. This speed is called the Terminal Velocity (\(v_T\)).
Did you know? A human skydiver usually reaches a terminal velocity of around \(55 \, \text{m\,s}^{-1}\) (about \(200 \, \text{km/h}\)). When they open their parachute, the parachute vastly increases the surface area, significantly increasing the drag force, causing the new terminal velocity to drop to a safe speed for landing!
3. Terminal Velocity Explained
The concept of terminal velocity is critical to this section of the syllabus. It is the steady speed achieved by an object falling through a fluid when the resistive force is equal to the driving force (usually weight).
3.1 Definition and Conditions
Definition: Terminal velocity (\(v_T\)) is the constant velocity reached by a falling object when the resultant force acting on it is zero (i.e., the drag force equals the weight).
The condition for terminal velocity:
\[F_{drag} = W\]
Since \(F_{net} = 0\), the object is in dynamic equilibrium—moving at a constant velocity.
Analogy: Think of a tug-of-war where both teams pull with exactly the same strength. The rope doesn't accelerate, but if it was already moving, it would continue moving at a steady pace.
3.2 Factors Affecting Terminal Velocity
Since terminal velocity is reached when \(F_{drag} = W\), anything that changes the weight or the efficiency of the drag force will change \(v_T\):
- Shape and Size: Objects with a larger cross-sectional area (like a spread-out parachute) experience a larger drag force at a given speed. Therefore, they reach a lower terminal velocity because less speed is required to make \(F_{drag} = W\).
- Mass/Weight: Heavier objects (\(W\) is larger) require a greater drag force to balance their weight. Since drag increases with speed, they must travel faster to achieve this balance, resulting in a higher terminal velocity. This is why a feather falls slower than a stone—the stone's weight is much larger compared to its drag area.
- Fluid Density: If the object falls in a denser fluid (like oil instead of air), the drag force will be greater for the same speed. This means the object reaches force balance (\(F_{drag} = W\)) at a lower speed, resulting in a lower terminal velocity.
Common Mistake to Avoid:
Students often say, "The acceleration becomes constant at terminal velocity." This is wrong! The acceleration becomes zero at terminal velocity. The velocity itself is constant.
3.3 Summary of the Falling Process (Step-by-Step)
When asked to describe the motion of an object falling with air resistance, use this structure:
- Initial Stage: \(v = 0\), \(F_{drag} = 0\). \(F_{net} = W\). Acceleration is \(g\) (maximum).
- Intermediate Stage: \(v\) increases, \(F_{drag}\) increases. \(F_{net}\) decreases. Acceleration decreases (but \(v\) is still increasing).
- Final Stage (Terminal Velocity): \(v\) reaches \(v_T\). \(F_{drag} = W\). \(F_{net} = 0\). Acceleration is zero. \(v\) is constant (maximum).
Key Takeaway from Non-Uniform Motion
Understanding non-uniform motion relies on linking the resultant force to Newton's Second Law (\(F_{net} = ma\)). Because the resistive force (drag) changes with speed, the net force is not constant, which means the acceleration is not constant. The concept of Terminal Velocity is the perfect example of dynamic equilibrium where the net force becomes zero, resulting in zero acceleration and a constant maximum speed.