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CBSE • कक्षा XI • Physics • अध्याय 3
अनुमानित समय: 45 Mins
प्रगति: अध्ययनरत

समतल में गति (Motion in a Plane)

In Class 11 Physics, "Motion in a Plane" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें
प्रस्तावना अध्याय अवधारणाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

⚽ Have You Ever Wondered?

When Lionel Messi curls a free-kick over a defensive wall into the top corner of a goal, what two independent motions combine into that graceful parab...

When Lionel Messi curls a free-kick over a defensive wall into the top corner of a goal, what two independent motions combine into that graceful parabolic arc? Projectile motion and vector mathematics govern two-dimensional trajectories.

यह अध्याय क्यों महत्वपूर्ण है

In Class 11 Physics, "Motion in a Plane" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्ययन से पूर्व (आवश्यक ज्ञान)

  • Vectors vs scalars from Class 9.
  • Trigonometric resolution.
  • Equations of motion.

इस अध्याय के लक्ष्य

  • Distinguish between Scalars and Vectors; define Unit vector, Position vector, and Zero vector.
  • Perform Vector Addition using Triangle Law and Parallelogram Law: $R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$.
  • Resolve a vector into rectangular components: $\vec{A} = A_x\hat{i} + A_y\hat{j}$.
  • Perform Vector Products: Dot product ($\vec{A} \cdot \vec{B} = AB\cos\theta$) and Cross product ($\vec{A} \times \vec{B} = AB\sin\theta\hat{n}$).
  • Analyze Projectile Motion: Trajectory equation (parabola), Time of flight ($T = \frac{2u\sin\theta}{g}$), Max height ($H = \frac{u^2\sin^2\theta}{2g}$), Horizontal Range ($R = \frac{u^2\sin 2\theta}{g}$, max at $45^\circ$).

अध्याय रूपरेखा एवं प्रगति

1 1. Vector Addition & Resolution
2 2. Scalar & Vector Products
3 3. Projectile Motion Mechanics

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Vector Addition & Resolution

The resultant of two vectors $\vec{A}$ and $\vec{B}$ inclined at angle $\theta$ is given by the Parallelogram Law: $$\mathbf{R = \sqrt{A^2 + B^2 + 2AB\cos\theta}} \quad \text{and} \quad \mathbf{\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}}$$ In Cartesian unit vectors: $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ where $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$.

2. Scalar & Vector Products

  • Dot Product (Scalar): $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z = AB\cos\theta$. (Perpendicular when $\vec{A} \cdot \vec{B} = 0$).
  • Cross Product (Vector): $\vec{A} \times \vec{B} = (AB\sin\theta)\hat{n}$. Evaluated via $3\times 3$ determinant. (Parallel when $\vec{A} \times \vec{B} = \vec{0}$).

3. Projectile Motion Mechanics

A body launched with velocity $u$ at angle $\theta$ moves with constant horizontal speed $u_x = u\cos\theta$ and vertical gravity $a_y = -g$:
• Trajectory Equation: $\mathbf{y = x\tan\theta - \frac{g x^2}{2u^2\cos^2\theta}}$ (Parabola!).
• Time of Flight: $\mathbf{T = \frac{2u\sin\theta}{g}}$.
• Maximum Height: $\mathbf{H = \frac{u^2\sin^2\theta}{2g}}$.
• Horizontal Range: $\mathbf{R = \frac{u^2\sin 2\theta}{g}}$ (Maximum when $\theta = 45^\circ$!).

चित्रात्मक व्याख्या एवं मॉडल

Motion in a Plane Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. Vector Addition & Resolution • 2. Scalar & Vector Products

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1
Parallelogram Law: Geometric vector addition computing resultant magnitude and direction.
मुख्य बिंदु 2
Dot Product: Scalar projection measuring directional alignment (zero when perpendicular).
मुख्य बिंदु 3
Cross Product: Pseudovector normal to the plane measuring rotational torque or area.
मुख्य बिंदु 4
Parabolic Trajectory: Quadratic ballistic curve resulting from gravity acting orthogonally to horizontal inertia.
मुख्य बिंदु 5
Complementary Launch Angles: Launch angles $\theta$ and $(90^\circ - \theta)$ achieve identical horizontal range.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

1
Show that the horizontal range of a projectile is the same for angles of projection $\theta$ and $(90^\circ - \theta)$.
उत्तर एवं व्याख्या देखें
उत्तर: $R(\theta) = \frac{u^2\sin 2\theta}{g}$. For $(90^\circ - \theta)$: $R' = \frac{u^2\sin 2(90^\circ - \theta)}{g} = \frac{u^2\sin(180^\circ - 2\theta)}{g} = \frac{u^2\sin 2\theta}{g} = R(\theta)$.
Sin(180 - 2θ) = Sin(2θ), ranges are identical.
2
Find the angle of projection for which the horizontal range and maximum height of a projectile are equal.
उत्तर एवं व्याख्या देखें
उत्तर: $R = H \implies \frac{u^2\sin 2\theta}{g} = \frac{u^2\sin^2\theta}{2g} \implies 2\sin\theta\cos\theta = \frac{\sin^2\theta}{2} \implies \tan\theta = 4 \implies \theta = \tan^{-1}(4) \approx 76^\circ$.
θ = tan^-1(4) ≈ 76°.
3
Find the unit vector in the direction of $\vec{A} = 3\hat{i} + 4\hat{j}$.
उत्तर एवं व्याख्या देखें
उत्तर: Magnitude $|\vec{A}| = \sqrt{3^2 + 4^2} = 5$. Unit vector $\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}$.
3/5 i + 4/5 j.
4
Determine whether the vectors $\vec{A} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{B} = 3\hat{i} + 2\hat{j}$ are perpendicular.
उत्तर एवं व्याख्या देखें
उत्तर: Compute dot product: $\vec{A} \cdot \vec{B} = (2)(3) + (-3)(2) + (1)(0) = 6 - 6 + 0 = 0$. Since their dot product is zero, the vectors are perpendicular.
Dot product is 0; vectors are perpendicular.
5
In uniform circular motion, why is there acceleration when the speed is constant?
उत्तर एवं व्याख्या देखें
उत्तर: Because velocity is a vector quantity; although speed (magnitude) is constant, the direction of motion continuously changes at every point, generating a centripetal acceleration ($a_c = v^2/r$) directed towards the center.
Direction of velocity changes continuously.
अध्याय का अध्ययन पूर्ण हुआ?
अभ्यास के लिए तैयार?

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कक्षा 11 Physics के सभी अध्याय

अध्याय 1: मात्रक और मापन (Units and Measurements) अध्याय 2: सरल रेखा में गति अध्याय 3: समतल में गति (Motion in a Plane) अध्याय 4: गति के नियम (Laws of Motion) अध्याय 5: कार्य, ऊर्जा और शक्ति अध्याय 6: कणों के निकाय तथा घूर्णी गति अध्याय 7: गुरुत्वाकर्षण (Gravitation) अध्याय 8: ठोसों के यांत्रिक गुण अध्याय 9: तरलों के यांत्रिकी गुण अध्याय 10: द्रव्य के तापीय गुण अध्याय 11: ऊष्मागतिकी (Thermodynamics) अध्याय 12: अणुगति सिद्धांत (Kinetic Theory) अध्याय 13: दोलन (Oscillations) अध्याय 14: तरंगें (Waves)

AI अध्ययन मित्र

त्वरित शंका समाधान

समतल में गति (Motion in a Plane) में कोई संदेह या प्रश्न है? हमारे AI अध्ययन मित्र से तुरंत समझें।

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