Welcome to the World of Changing Motion!
In our earlier studies, we often looked at objects moving with constant acceleration—like a ball falling in a vacuum where gravity is the only force. But in the real world, things are a bit "messier" (and more interesting!). Whether it’s a skydiver falling through the air or a marble dropping through a jar of honey, the acceleration often changes as the object moves. This is what we call motion with non-uniform acceleration.
By the end of these notes, you’ll understand why objects don't keep speeding up forever and how we can use graphs and experiments to track their journey.
1. Understanding Drag: The "Friendly" Resistance
Whenever an object moves through a fluid (which is just a fancy scientific word for any liquid or gas), it experiences a force that tries to slow it down. This force is called drag.
Key Definition: Drag is a frictional force experienced by an object travelling through a fluid. It always acts in the opposite direction to the object's motion.
What makes drag bigger or smaller?
Have you ever tried to run through waist-deep water? It’s much harder than running through air! That’s because several factors affect the size of the drag force:
- Speed: This is the big one. As an object's speed increases, the drag force acting on it also increases.
- Cross-sectional Area: A larger area pushing against the fluid means more drag. Think of an open umbrella versus a closed one.
- Shape: "Streamlined" or aerodynamic shapes (like a teardrop or a sports car) reduce drag.
- Fluid Density: Moving through a thick liquid (like oil) creates more drag than moving through a thin gas (like air).
Quick Review: Drag isn't a constant number. It grows as you go faster!
2. The Journey to Terminal Velocity
Don't worry if this seems tricky at first—let’s break down exactly what happens when an object (like a skydiver) falls through the air step-by-step.
Step 1: The Release
At the very moment the object is dropped, its speed is zero. This means there is no drag yet. The only force acting on it is its weight (\( W \)). Because the net force is at its maximum, the acceleration is at its maximum (\( a = g \)).
Step 2: Gaining Speed
As the object falls, it speeds up. Because it is moving faster, drag starts to increase. Now, the net force is the weight minus the drag (\( W - Drag \)). Since the net force is getting smaller, the acceleration starts to decrease. The object is still speeding up, just not as quickly as before.
Step 3: Reaching the Limit
Eventually, the object goes so fast that the Drag force grows to be exactly equal to the Weight. At this point, the forces are balanced.
\( Net \ Force = 0 \)
According to Newton’s Second Law (\( F = ma \)), if the net force is zero, the acceleration is zero. The object stops speeding up and stays at a steady speed.
Key Term: Terminal Velocity is the constant, maximum velocity reached by an object when the drag force is equal and opposite to the accelerating force (usually weight).
Key Takeaway: Acceleration is not constant here. It starts at \( 9.81 \ m \ s^{-2} \) and slowly drops to \( 0 \ m \ s^{-2} \) as terminal velocity is reached.
3. Visualizing Motion with Graphs
Since the acceleration is changing, our standard "SUVAT" equations won't work here! We have to rely on graphs to see what’s happening.
Velocity-Time (\( v-t \)) Graphs
For an object reaching terminal velocity, the line on a \( v-t \) graph is a curve that starts steep and flattens out into a horizontal line.
- Gradient: The gradient (slope) of the line represents the acceleration. Notice how the curve gets flatter? That shows the acceleration is decreasing.
- Horizontal line: When the line becomes perfectly flat, the gradient is zero, meaning the object has reached terminal velocity.
- Area under the graph: The area represents the displacement. Because the graph is a curve, you may need to estimate the area by counting the squares underneath the line.
Acceleration-Time (\( a-t \)) Graphs
This graph starts at a high value (like \( 9.81 \)) and curves downwards until it hits the zero line on the x-axis.
Did you know? A peregrine falcon reaches a much higher terminal velocity by tucking its wings in. This reduces its cross-sectional area and "streamlines" its shape to minimize drag!
4. Experimental Techniques (PAG 1)
In the lab, we often measure terminal velocity using two common methods. Here is how you can do it:
Method A: Ball-bearing in a Viscous Liquid
1. Fill a tall cylinder with a thick liquid like heavy oil or glycerin.
2. Drop a small steel ball-bearing into the liquid.
3. Use light gates or place rubber bands at equal intervals down the cylinder.
4. Record the time it takes for the ball to pass between each marker.
5. When the time intervals become constant, the ball has reached terminal velocity.
6. Calculation: \( v = \frac{distance \ between \ bands}{time \ taken} \).
Method B: Paper Cones in Air
1. Use a motion sensor connected to a data-logger, positioned above or below a falling paper cone.
2. The data-logger will automatically plot a \( v-t \) graph for you.
3. You can change the "mass" by nesting multiple cones inside each other (this increases weight without changing the shape/area significantly) to see how terminal velocity changes.
Quick Review: To find terminal velocity, you are looking for the point where velocity no longer changes with time.
5. Common Mistakes to Avoid
Mistake 1: Thinking acceleration is constant.
Correction: Acceleration is only constant if the net force is constant. Here, drag changes with speed, so the net force (and acceleration) changes too!
Correction: The acceleration is zero, but the velocity is at its maximum! The object is still moving very fast; it's just not getting any faster. Mistake 3: Confusing "Net Force" with "Drag".
Correction: Drag is just one force. The Net Force is the difference between Weight and Drag. At terminal velocity, Drag is large, but Net Force is zero.
Summary: The "Big Ideas"
- Drag is a resistance force that increases with speed, area, and fluid density.
- An object reaches terminal velocity when Weight = Drag.
- On a Velocity-Time graph, terminal velocity is shown by the graph leveling off to a horizontal line.
- Acceleration is the gradient of a \( v-t \) graph. For non-uniform acceleration, this gradient changes.
- You can estimate displacement by counting the squares (the area) under a curved \( v-t \) graph.