Motion with Variable Acceleration – Complete Notes, Revision, Important Questions & Downloads
Variable acceleration occurs when a(t), a(x), or a(v) is given instead of a constant. Three subtopics are covered: Acceleration as Function of Time (integrate a(t) to get v(t)); Acceleration as Function of Distance (use v dv/dx = a(x)); Acceleration as Function of Velocity (use dv/dt = a(v) or v dv/dx = a(v)). NEET rarely tests full integration — but it uses the concept in identifying which velocity-time relationship is non-linear, in checking whether motion is uniformly accelerated, and in slope-of-v-t-graph problems implying variable acceleration. Calculus-based manipulation of a = dv/dt and v = ds/dt is tested in JEE and less often in NEET, but the conceptual statements ('average speed equals instantaneous speed only at constant speed', 'distance never decreases') are standard NEET facts.
NEET Weightage — Motion with Variable Acceleration
Motion In One Dimension (Chapter 2)| NEET Year | Questions from this Topic | Bar | Marks |
|---|---|---|---|
| 2024 | 0 | 0 | |
| 2023 | 1 | 4 | |
| 2022 | 0 | 0 | |
| 2021 | 1 | 4 | |
| 2020 | 0 | 0 | |
| 2019 | 0 | 0 | |
| 6-Year Total (2019–2024) | 0–2 | 0–8 |
The conceptual rule 'distance never decreases with time' and 'average speed = instantaneous speed only at constant speed' are directly from the book's note section and have appeared as assertion-reason questions.
Full integral evaluations (∫a(t)dt) are more common in JEE. For NEET, the typical question gives v as a function of x or t and asks for acceleration at a specific instant.
How to Prepare Variable Acceleration for NEET
Focus on the v dv/dx technique The most common NEET variable-acceleration question gives v = f(x) and asks for acceleration. Use a = v(dv/dx) — differentiate v w.r.t. x, multiply by v. For v = kx: dv/dx = k, so a = kx × k = k²x. For v = k√x: dv/dx = k/(2√x), so a = (k√x)(k/2√x) = k²/2 (constant!). Recognise these patterns.
Know the conceptual statements from the textbook notes These statements are direct NEET material: '(1) Distance never decreases. (2) Average speed = instantaneous speed only at constant speed. (3) During translational motion, change in location.' Expect 1-mark questions based on these three statements.
Recognise when constant-a equations cannot be used If the v-t graph is curved (not linear), acceleration is variable. If x-t graph is a cubic or higher polynomial, a is variable. NEET tests: 'A body has velocity v = 3t² − 2t. Is it uniformly accelerated?' — a = dv/dt = 6t − 2 which changes with time, so NOT uniform.
Integration formulas — know the form, not the evaluation v = u + ∫₀ᵗ a(t)dt, x = ∫v dt, and for a(v): t = ∫dv/a(v), x = ∫v dv/a(v). Memorise these conceptually. NEET does not give complex integration limits — if integration appears, it is a simple polynomial.
Study Materials — Variable Acceleration
PDF · Cheat Sheet · MCQ Set · PYQSubtopics in Motion with Variable Acceleration
2-Column TableRapid Revision — Variable Acceleration
Concept → Trap → Example1) Acceleration as Function of Time
Concepta = f(t). Use a = dv/dt → v = u + ∫₀ᵗ f(t)dt. Then x = x₀ + ∫₀ᵗ v(t)dt. When a = constant, reduces to v = u + at (standard equation). Standard kinematics equations v=u+at are only valid for constant a.
- If a = kt: v = u + (1/2)kt²; x = x₀ + ut + (1/6)kt³. The motion is NOT uniformly accelerated (a increases with time).
- To determine if motion is uniformly accelerated: check if a = dv/dt = constant. If the v-t graph is linear, acceleration is constant. If v-t is curved, acceleration is variable.
- NEET trick: if a body has velocity v = 3t² + 2t + 1, its v-t graph is a parabola, meaning a = dv/dt = 6t + 2 is non-constant → variable acceleration.
2) Acceleration as Function of Distance
High Yielda = f(x). Use a = v(dv/dx) = f(x) → v dv = f(x)dx → integrate both sides: (v²−u²)/2 = ∫f(x)dx. Alternatively, 2∫f(x)dx = v² − u².
- Most common NEET case: v = kx. Then a = v(dv/dx) = kx × k = k²x. Acceleration is proportional to distance — this is simple harmonic motion-related.
- v = k√x: dv/dx = k/(2√x). a = (k√x)(k/2√x) = k²/2 = constant! Even though v varies, acceleration is constant.
- NEET question template: 'A particle moves such that its velocity is proportional to its displacement. Show that the acceleration is proportional to the displacement.' — directly tests a = v dv/dx for v = kx.
3) Acceleration as Function of Velocity
Concepta = f(v). Two forms: (1) a = dv/dt → t = ∫dv/f(v). (2) a = v(dv/dx) → x − x₀ = ∫v dv/f(v). Use form (1) for time questions, form (2) for distance questions.
- Form (1): dv/dt = f(v) → dt = dv/f(v) → t = ∫_{u}^{v} dv/f(v). This gives time as a function of velocity.
- Form (2): v dv/dx = f(v) → dx = v dv/f(v) → x = x₀ + ∫_{u}^{v} v dv/f(v). This gives position as a function of velocity.
- Drag-force scenario: if a = −kv (resistive force proportional to velocity), then t = ∫du/(-ku) → t = (1/k)ln(u/v). Used in fluid mechanics problems but rare in standard NEET.
US Curriculum Gaps — Variable Acceleration
Topics in this section are tested in NEET but covered differently in standard US physics courses.v(dv/dx) Technique (AP Physics 1 Gap)
AP Physics 1 (algebra-based) does not include the v dv/dx substitution for variable acceleration. NEET directly tests 'given v = f(x), find acceleration' using a = v(dv/dx) — this requires calculus. Students coming from AP Physics 1 background need to learn this technique specifically for the NEET problem type 'velocity as function of position.'
- AP Physics 1 only handles constant acceleration — kinematics equations of the form v = u + at
- The substitution a = v(dv/dx) is introduced in AP Physics C: Mechanics but not AP Physics 1
- NEET directly tests: 'v = kx, prove a ∝ x' — which requires this calculus technique
Conceptual Kinematics Statements (AP Physics C: Mechanics Gap)
While AP Physics C covers calculus kinematics numerically, it does not explicitly drill the conceptual statements that NEET tests in assertion-reason format: 'distance never decreases', 'average speed equals instantaneous speed only at constant speed', 'at constant velocity distance = displacement.' These are Indian textbook note-section statements used directly in NEET conceptual MCQs.
- AP Physics C covers kinematics rigorously but focuses on problem-solving, not statement-based MCQs
- NEET assertion-reason format requires exact knowledge of these TRUE/FALSE statements about distance vs displacement and speed
- The odd-number-ratio result (1:3:5 for free fall) and 'average = instantaneous only at constant speed' are NEET-specific drills
NEET-Style Practice Questions — Variable Acceleration
4 QuestionsPractice Problems — Variable Acceleration
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Physics — Motion In One Dimension Revision Checklist
Use this section for quick chapter tracking before mocks, part tests, and final NEET revision.
FAQ — Variable Acceleration
Notes · Downloads · Revision · Important QuestionsWhen can the standard equations v = u + at, s = ut + ½at² be used?
How do I find acceleration when velocity is given as a function of position?
Why does the formula a = v(dv/dx) work?
What does 'distance never decreases with time' mean physically?
When is average speed equal to instantaneous speed?
If v = kt (velocity proportional to time), what is the equation of motion?
If v = k√x (velocity proportional to √x), what is the acceleration?
Is variable acceleration tested numerically in NEET or only conceptually?
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