Unit 4 / SPPU FE Engineering Mechanics

Unit 4: Kinematics — SPPU FE Engineering Mechanics

Topics: Rectilinear motion, projectile motion, curvilinear motion, normal and tangential acceleration, and variable acceleration.

2019 pattern: end-sem Q5 / Q62024 pattern: end-sem Q5Most predictable unit in the course

Which exam is this in?

2019 pattern

End-sem Q5 / Q6, 18 marks, attempt 1 of the pair

2024 pattern

End-sem Q5: solve any 2 of 4 sub-questions, 12 marks

Unit-by-unit question frequency and what repeats

Unit IV — Kinematics

2024 pattern: Q5 (12 marks). 2019 pattern: Q5 / Q6.

Sub-topic	Frequency	Notes
Rectilinear: given x = f(t) or v = f(t), find velocity and acceleration	Every paper	Cubic position functions are recycled almost identically
Rectilinear: uniform acceleration (cars, trucks, braking)	Every paper
Projectile motion	Every paper	Golf ball at 20° or 25° is the single most recycled figure
Curvilinear: circular path, normal and tangential acceleration	Every paper	Truck on circular road appears identically in multiple years
Variable acceleration (integration, a = f(v) type)	Common	Hardest sub-type

Recurring problem families

Problem	Appeared in
Golf ball projected at 45 m/s at 20° — find max height and horizontal range	May/Jun 2023, Nov/Dec 2023, May/Jun 2025 (very similar form)
Golf ball projected at 50 m/s at 25° — find max height and horizontal range	May/Jun 2024, Nov/Dec 2023
x = t³ − 6t² + 9t + 5 — find velocity, acceleration, and time when v = 0	May/Jun 2025
x = t³ − 9t² + 12t + 5 — same type, different coefficients	Nov/Dec 2025
Truck on circular road, radius 50 m, aₜ = 0.05s m/s², moved s = 10 m — find total acceleration	May/Jun 2024, Apr 2025 (very similar form)
Runner on 126 m diameter track, speed increases from 4.2 to 7.2 m/s over 28.5 m — find total acceleration	May/Jun 2022, Nov/Dec 2025
Car decelerates from 80 km/h to rest in 30 m — find stopping distance from 110 km/h	May/Jun 2024, Nov/Dec 2023

Sub-topics and how often they appear

Sub-topic	Priority	Frequency
Rectilinear motion — position given as x = f(t), find velocity and acceleration	High	Every paper
Rectilinear motion — uniform acceleration (cars, trucks, braking distance)	High	Every paper
Projectile motion — max height and horizontal range	High	Every paper
Curvilinear motion — circular path, normal and tangential acceleration	Medium	Every paper
Variable acceleration — integration method, a = f(v) or a = f(x) type	Low	Common but hardest sub-topic, skip if time is short

🟢 High — do this first, appears in every paper.

🟡 Medium — do this if you have time.

🔴 Low — appears regularly but requires calculus confidence, skip if time is short.

Unit 4 has the most predictable questions in the course. The golf ball projectile, the cubic position function, and the car braking problem come back nearly every semester with the same numbers. Own the three high-priority topics and you have a reliable 8 to 12 marks available in this unit.

High

Rectilinear Motion — Position Given as x = f(t)

Formulas that appear

v = dx/dt

a = dv/dt = d²x / dt²

x = ∫v dt

a = v(dv/dx)

Set v = 0 to find when the particle momentarily stops or changes direction.

Set a = 0 to find when acceleration is zero, then substitute back into x and v.

Standard problem setup

A position function such as x = t³ − 6t² + 9t + 5 is given. The paper asks for velocity and acceleration at a specific time, and the time when velocity becomes zero.

1. Differentiate x to get v.

2. Differentiate v to get a.

3. Substitute the required time to get v and a at that instant.

4. Set v = 0 and solve the quadratic to get all valid times.

5. Substitute back into x if the question asks for position at those times.

The most recycled version is a cubic position function with changed coefficients but the same derivative workflow.

Common mistakes

Differentiating incorrectly. d/dt(t³) = 3t², not 3t³.

Forgetting the constant term disappears when differentiating position.

Only finding one root when v = 0 even though the quadratic has two valid times.

Confusing displacement with distance when the particle reverses direction.

High

Rectilinear Motion — Uniform Acceleration

Formulas that appear

v = u + at

s = ut + ½at²

v² = u² + 2as

s = ((u + v) / 2)t

Convert km/h to m/s by dividing by 3.6.

Convert m/s to km/h by multiplying by 3.6.

Standard problem setup

The most common question is a braking-distance problem: one speed-distance pair is given, and you must find stopping distance at a second speed using the same deceleration.

1. Convert the speeds from km/h to m/s first.

2. Use v² = u² + 2as with the first case to find deceleration a.

3. Reuse the same a in the second case to find the new stopping distance.

4. For overtaking questions, write position equations for both vehicles and set them equal.

Common mistakes

Not converting km/h to m/s before using the equations.

Treating deceleration as positive instead of writing it explicitly as negative.

Using the wrong kinematic equation for the unknowns given.

High

Projectile Motion

Formulas that appear

x = u cos θ × t

vₓ = u cos θ

vᵧ = u sin θ − gt

y = u sin θ × t − ½gt²

H = u² sin²θ / 2g

tᴴ = u sin θ / g

T = 2u sin θ / g

R = u² sin 2θ / g

Standard problem setup

A ball or golf ball is projected with a given speed and launch angle. The paper asks for maximum height and horizontal range, or sometimes for velocity at a given point.

1. Identify u, θ, and g = 9.81 m/s².

2. Use the height and range formulas directly when the projectile lands at the same level.

3. If velocity at a point is asked, find vₓ and vᵧ separately, then combine them.

4. If landing height is different, solve the vertical equation instead of using total time of flight.

The 45 m/s, 20° golf ball is one of the most repeated question families in the entire paper set.

Common mistakes

Reading sin 2θ as 2 × sin θ instead of sin(2θ).

Using g = 10 instead of 9.81 when the expected answers are based on 9.81.

Forgetting horizontal velocity stays constant throughout the flight.

Using T = 2u sin θ / g even when launch and landing heights are different.

Medium

Curvilinear Motion — Normal and Tangential Acceleration

Formulas that appear

aₜ = dv/dt

aₙ = v² / r

a = √(aₙ² + aₜ²)

Direction of total acceleration: tan⁻¹(aₙ / aₜ)

If aₜ = f(s), then aₜ = v(dv/ds)

Standard problem setup

The paper usually gives a truck or runner moving on a circular path with either a tangential acceleration function or enough data to find it from kinematics, then asks for total acceleration.

1. Find aₜ at the specified point.

2. Find the speed v at that point using the given relation or integration if needed.

3. Compute aₙ = v² / r.

4. Combine aₙ and aₜ to get total acceleration.

Common mistakes

Using diameter instead of radius in v² / r.

Confusing the inward normal direction with the tangential direction along the path.

Skipping integration when aₜ is given as a function of distance.

Low

Variable Acceleration — Integration Method

This is the hardest sub-topic in Unit 4. It appears regularly, but if your time is short you are better off locking the three high-priority topics before spending twenty minutes here.

Formulas that appear

When a = f(t): v = ∫a dt + C₁

Then x = ∫v dt + C₂

When a = f(v): dt = dv / a

When a = f(x): v(dv/dx) = a(x)

Standard problem setup

A particle has acceleration given as a function of time, velocity, or position. The key step is recognizing which integration form to use before writing any equation.

1. Identify whether a is given as a function of t, v, or x.

2. Rearrange into the correct differential form.

3. Integrate and apply the initial conditions to find the constants.

4. Substitute the requested value of t, v, or x to finish.

Common mistake

Using the uniform-acceleration equations when acceleration is not constant. If a depends on t, v, or x, you must integrate.

MCQ Sampler

3 free concept checks for Unit 4

These starter questions cover the highest-frequency ideas first. The full bank has 50 questions, and all 50 are temporarily open through June 30, 2026, including the harder hint layer.

Open full MCQ bank

Question 1

Velocity from Position

If displacement is given as x = f(t), the velocity function is obtained by:

Question 2

Acceleration from Velocity

If v = f(t), acceleration is found from:

Question 3

Momentarily at Rest

A particle is momentarily at rest when:

The remaining 47 questions are open right now. Use the beta window to work the harder concept checks before the paid boundary returns.

Sub-topics, formulas, setups, and mistakes on this page are drawn from 32 SPPU FE Engineering Mechanics papers from 2013 to 2026. See paper trends for the full frequency analysis.

Open practice drills