Physics (9630) | Projectile motion

Welcome to Projectile Motion!

Hi there! This chapter is all about understanding how things fly—like a basketball shot, a cricket ball, or even water squirting from a hose. It’s the physics of anything that is launched into the air and then moves freely under the influence of gravity alone.
Projectile motion might look complicated because it’s 2-dimensional (up/down and side-to-side), but we have a super-powerful secret weapon: we can treat those two directions completely independently! If you can master this separation, solving these problems becomes much easier. Let’s jump in!

3.2.4 The Core Concept: Independence of Motion

A projectile is any object moving through the air that is only acted upon by the force of gravity (in ideal cases, we ignore air resistance).

The most crucial concept in this chapter is the Principle of Independence of Motion:

The motion of a projectile in the vertical direction (up/down) is completely independent of its motion in the horizontal direction (side-to-side).

What does "Independent" really mean?

Imagine two people standing side-by-side. One person drops a ball straight down, and at the exact same moment, the second person throws an identical ball horizontally off a cliff edge.
Did you know? Both balls will hit the ground at the exact same time!
This happens because the downward acceleration due to gravity (\(g\)) affects them equally, regardless of how fast they are moving sideways.

Ideal Projectile Motion Assumptions (The Rules of the Game)

When solving most standard exam questions, we assume the motion is "ideal." This means we make two key assumptions:

1. Air resistance (drag) is negligible. This means there are no horizontal forces (except the tiny force applied at launch).
2. The acceleration due to gravity (\(g\)) is constant and acts only vertically downwards. This creates a uniform gravitational field. (We usually take \(g \approx 9.81 \, \text{m\,s}^{-2}\)).

Key Takeaway: We must solve 2D problems as two separate 1D problems: one horizontal, one vertical.

Analysis of Motion: Horizontal and Vertical

Motion in the Horizontal Direction (\(x\))

In the absence of air resistance, there are no forces acting horizontally.

• Acceleration \(a_x = 0\) (Zero acceleration).
• Velocity \(v_x\) is constant. The projectile maintains its initial horizontal speed throughout its flight.

Since the velocity is uniform (constant), we only need one simple formula:

Horizontal Distance: \(s_x = v_x t\)

This is easy! Half the work is done already.

Motion in the Vertical Direction (\(y\))

The motion vertically is always affected by gravity.

• Acceleration \(a_y = g\) (Uniform acceleration).
• Velocity \(v_y\) is changing. It decreases as the object rises, becomes zero at the maximum height, and increases as it falls.

Since the acceleration is uniform, we use the SUVAT equations.

Quick Review: The SUVAT Equations

Remember these four heroes from the motion in a straight line chapter:

1. \(v = u + at\)
2. \(s = \left( \frac{u+v}{2} \right) t\)
3. \(s = ut + \frac{1}{2}at^2\)
4. \(v^2 = u^2 + 2as\)

When applying these vertically, we replace \(s, u, v, a\) with \(s_y, u_y, v_y, a_y\). Since \(a_y\) is always \(g\), you can substitute \(a_y\) with \(\pm g\).

Important Note on Signs:
Consistency is key! You must define a positive direction (e.g., upwards).
If you choose Upwards as positive, then:

• Initial vertical velocity (\(u_y\)) is usually positive.
• Displacement (\(s_y\)) is positive if the final position is above the start.
• Acceleration due to gravity (\(a_y\)) is negative (\(-g\)) because gravity pulls downwards.

Projectiles Launched at an Angle

Most challenging problems involve launching an object at an angle \(\theta\) above the horizontal with an initial speed \(u\).

We must resolve the initial velocity \(u\) into its horizontal (\(u_x\)) and vertical (\(u_y\)) components using trigonometry:

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

Memory Aid: When resolving velocity, remember that the horizontal component (\(u_x\)) is along the adjacent side, so it uses COSINE. The vertical component (\(u_y\)) is opposite, so it uses SINE.

Step-by-Step Guide to Solving Projectile Problems

Don't worry if these look tricky! Follow these four steps and you can solve almost any ideal projectile problem:

1. Resolve and Define:
   • Draw a clear diagram.
   • Resolve the initial velocity \(u\) into components \(u_x\) and \(u_y\).
   • Define your positive direction (e.g., up is positive, right is positive).
2. List the knowns (SUVAT):
   Make two separate lists—one for Horizontal (\(x\)) and one for Vertical (\(y\)).
   Example list for a launch at 20 m/s at 30°:

   Horizontal:
   \(s_x = ?\)
   \(v_x = 20 \cos 30^\circ\)
   \(a_x = 0\)
   \(t = ?\)

   Vertical:
   \(s_y = ?\)
   \(u_y = 20 \sin 30^\circ\)
   \(v_y = ?\)
   \(a_y = -9.81 \, \text{m\,s}^{-2}\)
   \(t = ?\)

3. Find the Time (\(t\)):
   Time is the only quantity that is the same in both directions. You usually have to find the time first using the vertical SUVAT equations.
   Example: To find the total time of flight, set the final vertical displacement \(s_y\) back to zero.
4. Calculate the Unknown:
   Use the time calculated in step 3 to find the desired horizontal distance (range) or final vertical parameters (like final velocity).

Common Mistake to Avoid: DO NOT use the SUVAT equations with the total initial speed \(u\) and the total final distance \(s\). SUVAT only works in 1D along the line of acceleration. Projectile motion is 2D, so you MUST use components.

Projectiles in the Real World: Drag, Air Resistance, and Terminal Speed (Qualitative Treatment)

In reality, no object moves through the air without encountering air resistance, also known as drag force. The syllabus requires a qualitative understanding of these forces.

Air Resistance and Drag Force

• What is it? Drag is a force that opposes the motion of an object moving through a fluid (like air).
• How does it change? The drag force increases with speed. A faster object experiences much more drag.
• Effect on Trajectory:
  Drag reduces both the maximum height and the horizontal range of the projectile.
  It makes the trajectory asymmetrical. The descending part of the curve is steeper and shorter than the ascending part because the object is generally moving faster downwards than it was upwards (due to constant drag deceleration).

Lift Force

A lift force is another resistive force, usually acting perpendicular to the direction of motion, often generated by the shape of the object (like an aeroplane wing or the spin on a ball). We only need a qualitative treatment of this.

Terminal Speed (Maximum Speed)

When an object accelerates (like a car or a falling skydiver), the air resistance (drag) increases until it becomes equal in magnitude to the driving force (or weight, if falling).

Definition: Terminal speed (or terminal velocity) is the constant maximum velocity reached by a body when the net force acting on it is zero (i.e., driving forces = resistive forces).

The syllabus requires qualitative understanding of factors affecting the maximum speed of a vehicle:

• Maximum speed is reached when the engine's driving force is balanced by the total resistive forces (air resistance + friction).
• To increase maximum speed, you must either increase the engine power (driving force) or decrease the drag (e.g., by making the vehicle more aerodynamic).

Key Takeaway: Air resistance slows things down and makes trajectories uneven. Terminal speed is reached when forces balance out.