Physics · Kinematics · Chapter notes
Vectors and Kinematics · Class 11 Notes
Class 11 notes on vectors and kinematics: addition, dot and cross products, motion in a straight line, motion graphs and projectiles. The toolkit the rest of physics is written in. Free to read or download.
In short
Vectors are not a chapter you finish. They are the language every later chapter of physics is written in. A vector needs both size and direction, adds tip to tail, and splits into components along any two perpendicular directions you choose. Kinematics is the first place that language gets used in anger.
Contents
- ·How to Read This
- 1What a Vector Is and what it is not
- 2Adding Vectors triangle, parallelogram, components
- 3Dot and Cross Product a number and a vector
- 4Motion in a Straight Line the three equations and their limits
- 5Motion Graphs slope and area
- 6Projectile Motion two motions, one clock
- ★Vectors and Kinematics · Fact Sheet
How to Read This
- Get vectors right and Laws of Motion, Work and Energy, and Electrostatics all get easier.
- Blue marks a defining fact. Red marks a trap.
- Every formula here is derived from one idea: a vector can be split into pieces that act independently.
What a Vector Is
- A scalar has size only: mass, time, speed, work, energy.
- A vector has size and direction: displacement, velocity, force, momentum.
- Current has a direction but is a SCALAR, because it does not add by the triangle rule. That single fact is asked constantly.
- Equal vectors have the same magnitude and direction, wherever they are drawn.
- A unit vector has magnitude one and only carries direction.
Adding Vectors
- Triangle law: place B's tail on A's head, and R closes the triangle.
- Parallelogram law gives the same answer with both tails together.
- Magnitude: R = √(A² + B² + 2AB cosθ).
- Vectors never add like numbers. Two forces of 3 N and 4 N can give anything from 1 N to 7 N.
- To subtract, reverse one vector and add.
- Split every vector into x and y parts.
- Then add all the x parts, and separately all the y parts.
- This turns a geometry problem into arithmetic.
- It is how every two-dimensional problem in physics is actually solved.
Dot and Cross Product
- Dot product: A · B = AB cosθ. The result is a scalar.
- It is zero when the vectors are perpendicular. Work is a dot product, which is why a force at 90° does none.
- Cross product: |A × B| = AB sinθ. The result is a vector, perpendicular to both.
- It is zero when the vectors are parallel. Torque and angular momentum are cross products.
- A × B = −(B × A), so the order matters. The dot product does not care about order.
Motion in a Straight Line
- Distance is the path length and is a scalar. Displacement is the straight line from start to finish.
- So average speed is distance over time, but average velocity is displacement over time.
- They are not the same. Walk a full circle and your average velocity is zero.
- The three equations: v = u + at, s = ut + ½at², and v² = u² + 2as.
- All three assume constant acceleration. If a changes, you must integrate instead.
- Throw a ball straight up. At the very top its velocity is zero.
- But gravity has not switched off, so the acceleration is still g downward.
- That is exactly why it comes back down.
- This is the most common conceptual question in kinematics.
Motion Graphs
- Slope of x against t gives velocity.
- Slope of v against t gives acceleration.
- Area under v against t gives displacement, and area under a against t gives change in velocity.
- Area below the axis is negative, so it subtracts. Total distance needs the sizes added instead.
Projectile Motion
- Horizontal and vertical motions are completely independent. They only share the clock.
- Horizontally there is no acceleration, so the horizontal velocity never changes.
- Vertically it is free fall, with acceleration g downward throughout.
- Time of flight T = 2u sinθ / g.
- Maximum height H = u²sin²θ / 2g.
- Range R = u²sin2θ / g, which is largest at θ = 45°.
- Complementary angles give the same range, so 30° and 60° land in the same place.
- At the top the velocity is not zero. Only its vertical part is. The horizontal part continues.
★ Vectors and Kinematics · Fact Sheet
Every rule for revision day.
SCALAR OR VECTOR
Vector: displacement, velocity,force, momentum
Current has direction but is SCALAR.
ADDING
R = √(A² + B² + 2AB cosθ)θ=0 maximum, θ=180° minimum
3 N and 4 N give 1 N to 7 N.
DOT PRODUCT
A · B = AB cosθa SCALAR, zero when perpendicular
Work is a dot product.
CROSS PRODUCT
|A × B| = AB sinθa VECTOR, zero when parallel
A × B = −(B × A).
THE THREE EQUATIONS
v = u + ats = ut + ½at²
v² = u² + 2as.
THEIR LIMIT
All three need CONSTANT a.If a changes, integrate.
GRAPHS
x-t slope = velocityv-t slope = acceleration
v-t area = displacement.
PROJECTILE
T = 2u sinθ/gH = u²sin²θ/2g
R = u²sin2θ/g, max at 45°.
PROJECTILE TRAPS
At the top v is NOT zero.Complementary angles
give the same range.
Now play it.
Reading a chapter and understanding it are different things. In the app this chapter becomes a game you play, a short read, then practice that tests you at every step. A full chapter, understood, in 45 to 60 minutes. Free to start.
- PlayA game built for the concept
- ReadThe short NCERT explainer
- PracticeTimed, until it sticks
Still stuck at 11pm? TarQPro reads the answer you got wrong and teaches the idea behind it, not just the right option.