Unit 21
Mechanics
Class 12 Mathematics
Dynamics
Class 12 Mathematics – Dynamics Notes PDF
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This chapter develops dynamics through motion of a particle, Newton’s laws of motion, and projectile motion. The emphasis is on mathematical modeling and solving problems.
1. Motion of a Particle in a Straight Line
If displacement is s(t),
For constant acceleration:
2. Newton’s Laws of Motion
A body remains at rest or in uniform straight-line motion unless acted on by a resultant external force.
Resultant force equals the rate of change of momentum; for constant mass:
F=ma.Interaction forces between two bodies are equal in magnitude and opposite in direction.
3. Momentum and Impulse
For constant force over time interval Δt, J=FΔt.
4. Projectile Motion
A projectile is modeled as a particle moving under gravity alone after projection, neglecting air resistance.
If it is projected with speed u at angle θ:
Horizontal acceleration is zero; vertical acceleration is −g.
5. Equation of the Trajectory
Eliminating t:
This is a quadratic equation in x, so the trajectory is a parabola.
6. Time of Flight, Maximum Height and Range
When a projectile lands at the same level from which it is launched:
Maximum range for fixed u occurs at θ=45°, giving Rmax=u²/g.
7. Worked Examples
If s=t³−3t²+2t, then v=3t²−6t+2 and a=6t−6.
A particle is projected at 20 m/s at 30°. Taking g=9.8, T=2(20)(1/2)/9.8≈2.04 s.
8. Common Mistakes
Keep the sign of gravitational acceleration consistent.
Do not use same-level range/time formulae when launch and landing heights differ.
Resolve velocity, not acceleration, into initial horizontal/vertical components.
Use the net/resultant force in F=ma.
9. Important Exam Questions
- Find velocity and acceleration from a displacement function.
- Apply Newton’s second law to a particle.
- Derive the projectile trajectory equation.
- Derive time of flight, maximum height and range.
- Show that maximum range occurs at 45°.
- Solve projectile numericals.
Discussion
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