Introduction to Projectile Motion

Imagine you kick a football or throw a stone into the air. It doesn't just go straight up and down; it follows a curved path through the air. This type of movement is called projectile motion. In this chapter, we will learn how to predict exactly where that object will land and how high it will go.

The secret to understanding projectiles is simpler than it looks: we just treat the movement sideways and the movement upwards as two completely separate events happening at the same time. Don't worry if this seems a bit strange at first—once you see how they work independently, the math becomes much easier!

Note: This chapter builds on your knowledge from Motion along a straight line. If you need a refresher on the basic SUVAT equations, you might want to look back at that section briefly.

The Golden Rule: Independence of Motion

The most important thing to remember in this entire chapter is the independence of vertical and horizontal motion.

  • Horizontal Motion: Because we usually ignore air resistance in basic problems, there is no force acting horizontally. This means the horizontal velocity is constant.
  • Vertical Motion: The force of gravity is always pulling the object down. This means the object will accelerate downwards at a constant rate of \(g\) (\(9.81 \, \text{m s}^{-2}\)).

Analogy: Imagine two balls. One is dropped vertically, and the other is fired horizontally at the exact same time. Even though one is moving sideways, they will both hit the ground at the exact same moment because their vertical journeys are identical!

Quick Review: The "Two-World" View

When solving a problem, always split your page into two columns:

Horizontal (\(x\)): Velocity (\(v\)) never changes. Use \(s = vt\).
Vertical (\(y\)): Velocity changes due to gravity. Use SUVAT equations.

Step 1: Resolving the Initial Velocity

Most projectiles are launched at an angle (\(\theta\)) with an initial velocity (\(u\)). Before you can do anything else, you must split this into its horizontal and vertical components using trigonometry.

Horizontal component (\(u_x\)): \(u_x = u \cos(\theta)\)
Vertical component (\(u_y\)): \(u_y = u \sin(\theta)\)

Memory Trick: To find the horizontal side, you go "across" the angle (Cos is "across"). To find the vertical side, you go "opposite" the angle (Sin is "opposite").

Step 2: Using the Equations of Motion

Once you have your components, you can use the equations for uniform acceleration (SUVAT). Remember, the only thing that links the horizontal and vertical sides is time (\(t\)). Time is the same for both!

Common Vertical SUVAT values:
  • \(s\) = vertical displacement (height)
  • \(u\) = \(u \sin(\theta)\)
  • \(v\) = final vertical velocity (at the peak of flight, \(v = 0\))
  • \(a\) = \(-g\) (usually \(-9.81 \, \text{m s}^{-2}\))
  • \(t\) = time
Common Horizontal values:
  • \(s\) = horizontal range
  • \(v\) = \(u \cos(\theta)\) (this stays constant)
  • \(t\) = time

Key Takeaway: If you are stuck, try to find the time (\(t\)) using the vertical information first. You can then use that time to find the horizontal distance.

Real-World Factors: Friction, Drag, and Lift

In the "perfect" world of Physics problems, we often ignore air resistance. However, the AQA syllabus requires you to understand qualitatively (in words, not math) what happens when we include it.

1. Air Resistance (Drag and Friction)

As an object moves through the air, it collides with air molecules. This creates a force called drag (or air friction) that always acts in the opposite direction to the motion.

Effect on the path:

  • The horizontal velocity decreases over time.
  • The maximum height reached is lower.
  • The range (distance travelled) is shorter.
  • The path is no longer a perfect parabola; it becomes steeper on the way down.

2. Lift

Some objects (like a frisbee or a wing) experience lift. This is an upwards force caused by the object's shape and its motion through the air. Lift can counteract gravity, allowing the object to stay in the air longer and travel further.

3. Terminal Speed

As an object falls faster, the drag force acting upwards increases. Eventually, the drag force becomes equal in size to the weight of the object acting downwards.

When Drag = Weight:

  • The resultant force is zero.
  • The acceleration is zero.
  • The object moves at a constant speed called terminal speed.

Did you know? A skydiver's terminal speed is roughly \(120 \, \text{mph}\), but if they pull their parachute, the drag increases massively, slowing their terminal speed down to a safe landing pace!

Summary Checklist

- Can you split a velocity vector into horizontal (\(\cos\)) and vertical (\(\sin\)) parts?
- Do you remember that horizontal velocity is constant (if air resistance is ignored)?
- Do you know that vertical motion is always accelerating at \(9.81 \, \text{m s}^{-2}\) downwards?
- Can you explain how drag makes the projectile's path shorter and less symmetrical?
- Do you understand that terminal speed happens when drag equals weight?

Common Mistakes to Avoid:
  • Mixing up components: Never use the horizontal velocity in a vertical SUVAT equation!
  • Forgetting the sign of \(g\): If you decide "up" is positive, then gravity (\(g\)) must be negative (\(-9.81\)) because it acts downwards.
  • Time at the top: If you calculate the time taken to reach the maximum height, remember you usually need to double it to find the total time of flight (if it lands at the same height it started).