IP Physics Notes (Upper Secondary, Year 3-4): 2) Kinematics

Quick recap -- Recognise scalar vs vector motion, read gradients/areas off kinematics graphs, treat free fall as constant acceleration, and deploy the correct SUVAT equation based on the knowns you have.

Scalars, Vectors & Core Definitions

• Distance (scalar) vs displacement (vector) -- displacement carries direction.
• Speed (scalar) vs velocity (vector) -- velocity specifies direction.
• Acceleration: rate of change of velocity, \( a = \dfrac{v - u}{t} \).
• Average speed \( = \dfrac{d}{t} \) (distance travelled \(d\) over elapsed time \(t\)); average velocity \( = \frac{\Delta s}{t} \) with \(\Delta s\) measured as displacement.

Graphical Analysis

Displacement-Time (s-t)

• Gradient = velocity.
• Straight line -> uniform velocity; curve getting steeper -> acceleration; horizontal line -> at rest.

Velocity-Time (v-t)

• Gradient = acceleration. Area under the curve = displacement.
• Horizontal line -> uniform velocity; sloping line -> uniform acceleration or deceleration; crossing the axis -> change in direction.

Typical motions:

1. Constant velocity -> horizontal line above time axis.
2. Constant acceleration -> straight line rising from the origin.
3. Constant deceleration -> straight line falling towards zero.
4. At rest -> line on the time axis.
5. Changing acceleration -> curved or triangular profile.
6. Negative velocity -> line below the axis; slope sign tells you about acceleration.

Acceleration-Time (a-t)

• Horizontal segments show constant acceleration/deceleration; area under the curve gives change in velocity.
• Step changes capture sudden throttle/braking events.

Free Fall Essentials

• Treat gravity as uniform acceleration downward: use \( g = \pu{9.81 m.s-2} \) (or \( \pu{10 m.s-2} \) for quick estimates).
• Dropped object: \( u = 0 \), acceleration downward. Thrown upward: \( u > 0 \) upwards, acceleration still downward (negative in your coordinate system).
• Time up \( = \frac{u}{g} \), maximum height \( = \frac{u^2}{2g} \), total flight time \( = \frac{2u}{g} \) if landing at launch height.

SUVAT Toolkit

List the known quantities among \( s, u, v, a, t \). Strike out what you lack, then choose the single SUVAT equation that avoids the missing variable. No simultaneous equations needed.

Core formulas (uniform acceleration):

• \( v = u + at \)
• \( s = ut + \tfrac{1}{2} a t^2 \)
• \( v^2 = u^2 + 2as \)
• \( s = \tfrac{1}{2} (u + v) t \)
• \( s = vt - \tfrac{1}{2} a t^2 \)

Worked Examples

Two-phase car -- Brake from \( \pu{25 m.s-1} \) to \( \pu{10 m.s-1} \) with \( a = -\pu{4 m.s-2} \), then accelerate at \( +\pu{2 m.s-2} \) for \( \pu{12 s} \).

1. Braking time: \( v = u + at \Rightarrow 10 = 25 + (-4)t \Rightarrow t = \pu{3.75 s} \).
2. Braking distance: \( s = ut + \tfrac{1}{2}at^2 \Rightarrow s = 25(3.75) + \tfrac{1}{2}(-4)(3.75)^2 = \pu{65.6 m} \).
3. Second phase distance: \( s = ut + \tfrac{1}{2}at^2 \) with \( u = \pu{10 m.s-1} \), \( a = \pu{2 m.s-2} \), \( t = \pu{12 s} \) -> \( s = \pu{264 m} \).
4. Final speed after second phase: \( v = u + at = 10 + 2 \times 12 = \pu{34 m.s-1} \).

Projectile example -- Start from a 50 m cliff, launch at 30 m/s and 40 degrees above the horizontal.

Split components:

• \( u_x = 30 \cos 40^{\circ} \)
• \( u_y = 30 \sin 40^{\circ} \)

Vertical displacement to ground: \( s = -\pu{50 m} \), \( u = u_y \), \( a = -\pu{9.81 m.s-2} \).

Use \( s = ut + \tfrac{1}{2}at^2 \) -> quadratic in \( t \). Solve for the positive root to get flight time \( t \approx \pu{5.2 s} \).

Horizontal range: \( x = u_x t \) (no horizontal acceleration) -> \( x \approx 23 \times 5.2 \approx \pu{120 m} \).

Key Takeaways

• Interpret gradients and areas across s-t, v-t, a-t graphs to move fluently between displacement, velocity, and acceleration.
• Free fall simplifies to constant downward acceleration; watch sign conventions when an object reverses direction.
• SUVAT equations remain the fastest route for uniform-acceleration problems--always identify which variable is absent before picking an equation.