Introduction to Motion with Non-Uniform Acceleration
Welcome! So far in your Physics journey, you’ve likely spent a lot of time working with constant acceleration (like those trusty SUVAT equations). However, in the real world, acceleration rarely stays the same. Whether it’s a skydiver jumping from a plane or a marble dropping into a jar of honey, the forces involved change as the object moves. In this chapter, we will explore why this happens and how we can describe this "non-uniform" motion.
1. Understanding Drag
When an object moves through a fluid (which is just a fancy Physics word for any liquid or gas), it experiences a resistance force called drag.
What is Drag?
Drag is a frictional force that acts in the opposite direction to the motion of an object. If you are running forward, drag is pushing you backward. It’s the fluid’s way of trying to slow you down!
Factors Affecting Drag
Why are some objects "dragged" back more than others? There are four main factors you need to know:
- Speed: This is the big one. As an object moves faster, it has to push more fluid particles out of the way every second. In most cases, drag is proportional to the square of the speed: \( Drag \propto v^2 \).
- Cross-Sectional Area: The "wider" the object, the more fluid it hits. Think of walking through a swimming pool with a large piece of plywood versus walking through it normally.
- Shape (Aerodynamics): Streamlined shapes (like a teardrop or a sports car) allow fluid to flow past easily, reducing drag.
- Viscosity of the Fluid: This describes how "thick" the fluid is. It's much harder to move through treacle than it is to move through air!
Quick Tip: Remember that drag always increases as speed increases. This is the "engine" behind non-uniform acceleration.
2. Objects Falling with Drag
When an object falls through the air, it isn't just affected by its weight. The net force (the overall force) determines its acceleration. We use Newton's Second Law for this: \( F_{net} = ma \).
The Step-by-Step Process of Falling
Imagine dropping a ball from a high building. Let's look at what happens at three different stages:
Stage 1: The Moment of Release
At the very start, the velocity is zero. Because \( v = 0 \), the drag is zero. The only force acting on the ball is its weight. Therefore, the acceleration is at its maximum (roughly \( 9.81 \text{ m s}^{-2} \)).
Stage 2: Speeding Up
As the ball falls, it gets faster. Because it is faster, the drag force increases. Now, the net force is the Weight minus the Drag. Since the Weight stays the same but the Drag is growing, the net force decreases. This means the acceleration decreases. The ball is still speeding up, but it's speeding up more slowly than before.
Stage 3: Equilibrium (No more acceleration)
Eventually, the ball reaches a speed where the drag force is exactly equal to the weight. The forces are now balanced. The net force is zero, so the acceleration is zero. The ball has reached its maximum possible speed.
Key Takeaway: In the presence of drag, acceleration starts at \( g \), decreases over time, and eventually becomes zero.
3. Terminal Velocity
The constant, maximum speed reached by an object when the drag force equals the accelerating force (usually weight) is called terminal velocity.
Did you know? A human skydiver usually has a terminal velocity of about 120 mph (54 m/s) in a "belly-to-earth" position. If they dive head-first, they reduce their cross-sectional area, which reduces drag and increases their terminal velocity to over 200 mph!
Visualizing Terminal Velocity
On a Velocity-Time graph for an object falling with drag:
- The line starts steep (gradient = \( g \)).
- The curve starts to flatten out as acceleration decreases.
- The line becomes a horizontal straight line when terminal velocity is reached.
Common Mistake: Many students think that when an object reaches terminal velocity, it stops moving. It doesn't! It just stops accelerating. It continues to fall at a steady, fast speed.
4. Investigating Terminal Velocity (PAG 1)
You may be asked how to determine the terminal velocity of an object in a lab setting. A common experiment involves dropping a small ball-bearing into a tall cylinder of a viscous liquid (like heavy oil or glycerol).
Step-by-Step Procedure:
- Fill a tall graduated cylinder with a viscous liquid.
- Mark "test intervals" at regular distances down the cylinder using elastic bands.
- Drop the ball-bearing into the liquid.
- Use a stopwatch to record the time it takes to pass each marker, or better yet, use light gates connected to a data-logger for better precision.
- Calculate the velocity between each marker using \( v = \frac{\Delta s}{\Delta t} \).
- When the velocity becomes constant for several intervals, you have found the terminal velocity.
Alternative Method: You can use video analysis (filming the drop and using software to track the position frame-by-frame). This is often more accurate than using a stopwatch because it eliminates human reaction time errors.
Summary Review Box
Concept Check:
1. Drag increases with speed and area.
2. Acceleration decreases as drag increases.
3. Terminal Velocity happens when Weight = Drag.
4. At terminal velocity, Net Force = 0 and Acceleration = 0.
Don't worry if the idea of "decreasing acceleration" feels weird! Just remember: acceleration is the "rate" of speeding up. If acceleration is decreasing, you are still getting faster, you're just not gaining speed as quickly as you were a second ago.