Motion with resistive forces
Section: Dynamics | Syllabus: Cambridge AS Level Physics (9702)
Friction
Friction is a resistive force that acts where surfaces contact each other. It always acts in the opposite direction to the motion or the intended motion.
- Translational Equilibrium: According to Newton's First Law, an object in motion will continue with uniform velocity unless a resultant force acts. On a solid surface, friction is usually that force.
- Work Done: Friction converts kinetic energy into thermal energy (heat).
If a driving force exceeds the friction force, the object accelerates. The friction reduces the net acceleration: a = F_driving - F_frictionm.
Drag and Viscosity
Objects moving through fluids (liquids or gases) experience a resistive force called drag (or viscous drag).
Factors Affecting Drag:
- Speed: Greater speed leads to greater drag. For many objects, drag is proportional to v^2.
- Cross-sectional Area: A larger area presented to the fluid increases drag because more fluid must be pushed aside.
- Shape (Streamlining): Streamlined shapes allow smooth, non-turbulent flow, significantly reducing drag compared to blunt shapes.
- Viscosity: This is the "thickness" or resistance of a fluid to flow. Air has low viscosity; water has higher viscosity; oil or honey is very viscous.
Two objects of the same size and shape moving at the same speed through the same fluid experience the same drag force, regardless of their mass.
Terminal Velocity
When an object falls under gravity, it eventually reaches a constant speed called terminal velocity (v_t), where the upward drag equals the downward weight.
The Process of Reaching v_t:
- Release (t=0): v=0, so Drag (D) = 0. Resultant force = W. Acceleration = g.
- Falling (v increasing): D increases. Net force (W - D) decreases. Acceleration decreases.
- Equilibrium (D = W): Net force = 0. Acceleration = 0. The object continues at a constant terminal velocity.
Show a sequence of three free-body diagrams for a falling ball: (1) At t=0, only weight arrow. (2) Mid-fall, weight arrow and small drag arrow. (3) Terminal velocity, weight and drag arrows are equal length.
The Role of Mass (Table Tennis vs. Golf Ball)A golf ball and a table tennis ball have equal diameters but different masses (m_golf approx 20 imes m_tt).
ExplanationAt the same speed, both experience the same drag force. However, because a = racW - Dm, the lighter table tennis ball experiences a much larger deceleration effect from the resistance. It reaches terminal velocity sooner and at a lower speed. The heavier golf ball continues accelerating for longer.
Worked Example: Falling in Different FluidsThree identical steel balls are dropped: A (vacuum), B (air), C (water).
Analysis- Ball A (Vacuum): No resistance. v increases uniformly (a = g). The v-t graph is a straight line through the origin.
- Ball B (Air): Low resistance. Acceleration decreases slightly over 1m. Graph curves slightly.
- Ball C (Water): High resistance (viscosity). Acceleration decreases significantly. v increases much less rapidly than in air.
Plot velocity vs time for a vacuum (diagonal straight line), air (curve leveling off at high speed), and water (curve leveling off quickly at low speed).
Horizontal Motion and Maximum Speed
Vehicles reach a maximum speed when the resistive forces (air resistance + friction) become equal to the maximum driving force provided by the engine.
Worked Example: Thrust SSC Land Speed RecordWhy must a land speed record involve two passes in opposite directions within one hour?
SolutionThis accounts for wind speed and ground slope. Running in both directions ensures that any advantage from a tailwind or downhill slope is cancelled out when results are averaged.
Projectile Motion with Air Resistance
In real-world conditions, air resistance acts in the direction opposite to the velocity vector at every instant.
Effects of Air Resistance:- Reduced Range: The horizontal component of velocity decreases over time.
- Reduced Height: The vertical upward motion is opposed by both gravity and air resistance.
- Asymmetry: The path is no longer a perfect parabola. It is "steeper" on the way down than on the way up.
Overlay two projectile paths: a dashed symmetrical parabola (no air resistance) and a solid asymmetrical path (with air resistance) that has a lower peak and shorter range.
Key Definitions
- Terminal Velocity: Constant speed reached when Drag = Weight.
- Viscosity: Resistance of a fluid to flow.
- Streamlining: Designing shapes to minimize turbulent flow and drag.
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