4.1 Linear Motion & Motion Under Gravity

Linear Motion & Motion Under Gravity

Kinetics (in the Part-66 syllabus sense of Mechanics — Kinetics) studies how bodies move: how far, how fast, and how their speed changes with time. Chapter 3 covered forces and equilibrium when things stay put; this section treats rectilinear (straight-line) motion with constant acceleration, including free fall. Dynamics (Newton’s laws linking force to acceleration) follows in the next chapter — here the focus is the kinematic description itself.

Distance, Displacement, Speed, and Velocity

Distance is the total path length travelled. It is always positive and is a scalar. Displacement is the straight-line change of position from start to finish, with direction — a vector. Walking 10 m forward and 10 m back gives distance 20 m but displacement 0.

Speed is the rate of covering distance:

speed = distance / time

Units: metres per second (m/s). Average speed uses total distance over total time; instantaneous speed is the limit as the time interval shrinks.

Velocity is the rate of change of displacement:

velocity = displacement / time

Velocity is a vector — magnitude and direction. If an aircraft taxis 200 m north in 40 s, its average velocity is 5 m/s north. The average speed is also 5 m/s if it went only north; reverse or turn, and speed and the magnitude of average velocity diverge.

Exam trap: a quantity without direction stated is often speed; “20 m/s east” is velocity. Average velocity over a closed path can be zero while average speed is not.

Acceleration

Acceleration is the rate of change of velocity:

a = (v − u) / t  or  a = Δv / Δt

where u is initial velocity, v is final velocity, and t is the time interval. SI unit: m/s². Acceleration is a vector. Speeding up in the direction of travel is positive acceleration in that sense; slowing down (deceleration) is acceleration opposite the velocity. A body can have constant speed and still accelerate if direction changes (circular motion — next section).

Uniform (constant) acceleration means a does not change with time. That idealisation is excellent for many Module 2 problems: braking on a runway approximation, free fall without air resistance, and constant-thrust start rolls treated as constant a.

Worked example — taxi acceleration. An aircraft starts from rest (u = 0) and reaches 30 m/s in 20 s along a straight taxiway. Acceleration is:

a = (30 − 0) / 20 = 1.5 m/s²

If it later brakes from 30 m/s to rest in 15 s:

a = (0 − 30) / 15 = −2.0 m/s²

(The negative sign means opposite the positive-forward direction.)

Equations of Motion (Constant Acceleration)

When acceleration is constant, three core equations relate u, v, a, t, and displacement s:

1. v = u + at
2. s = ut + ½at²
3. v² = u² + 2as

(A fourth form, s = ½(u + v)t, follows from average velocity under constant a, but the three above cover nearly all exam work.)

How to choose which equation. List the knowns and the unknown. Use (1) when you need v or t and s is missing. Use (2) when s is required and v is missing. Use (3) when t is missing and you know or need s with speeds. Always fix a sign convention first: pick a positive direction (e.g. upward, or along the runway) and keep u, v, a, and s consistent with it.

Worked example — distance under constant acceleration. A trolley on a hangar floor accelerates from rest at 0.8 m/s² for 5.0 s. Distance travelled:

s = ut + ½at² = 0 + ½(0.8)(5.0)² = 0.4 × 25 = 10 m

Final speed: v = 0 + 0.8 × 5 = 4.0 m/s. Check with (3): v² = 0 + 2(0.8)(10) = 16 ⇒ v = 4.0 m/s. Consistent.

Worked example — stopping distance. A vehicle approaches a stop mark at 12 m/s and brakes with constant deceleration of magnitude 3.0 m/s². Take forward as positive: u = 12 m/s, v = 0, a = −3.0 m/s². From (3):

0 = 12² + 2(−3.0)s ⇒ 0 = 144 − 6s ⇒ s = 144/6 = 24 m

Time to stop from (1): 0 = 12 + (−3)t ⇒ t = 4.0 s.

Motion Under Gravity (Free Fall)

Near Earth’s surface, if air resistance is neglected, every freely falling body has the same downward acceleration g. Standard value:

g ≈ 9.81 m/s²  (often 9.8 m/s² or 10 m/s² for quick estimates)

“Freely falling” includes objects thrown upward: after release, the only acceleration is still g downward until impact or catch. At the highest point, instantaneous velocity is zero, but acceleration remains g downward — a classic misconception trap.

Sign convention tip. Two common choices:

• Positive downward: for a drop from rest, u = 0, a = +g, s positive down.
• Positive upward: for a throw upward, u positive, a = −g, and displacement at max height is positive s with v = 0.

Either works if applied consistently.

Worked example — dropped tool. A spanner is dropped from rest from a stand 4.905 m above the hangar floor. Take down positive, g = 9.81 m/s²:

v² = u² + 2as = 0 + 2(9.81)(4.905) = 96.26 ⇒ v ≈ 9.81 m/s

t from s = ½gt²: 4.905 = ½(9.81)t² ⇒ t² = 1 ⇒ t = 1.0 s

Impact speed about 9.8 m/s after one second of free fall from rest matches the rule of thumb that speed increases by roughly g each second when starting from rest and falling freely.

Worked example — projected upward. A marker is thrown vertically upward at 15 m/s. Take up positive, a = −9.81 m/s². Maximum height (v = 0):

0 = 15² + 2(−9.81)s ⇒ s = 225 / 19.62 ≈ 11.5 m

Time to apex: 0 = 15 − 9.81t ⇒ t ≈ 1.53 s. Total flight time to return to the throw height is twice that (symmetric if landing at same level), about 3.06 s. Speed on return equals launch speed, direction reversed.

Aviation Context

Straight-line kinematics models take-off and landing ground rolls when acceleration is treated as roughly constant, fall times for objects released from height, and relative motion along a single axis (e.g. a part sliding on a straight rail). Real aircraft accelerate under varying thrust and drag — Module 2 expects the ideal constant-a toolkit, not full performance charts. Always state assumptions: constant a, neglect of air resistance for free fall, and SI units throughout.

Quantity Summary

Master the definitions, the three equations, and free-fall sign discipline — that combination answers the large majority of linear-motion Module 2 items.