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Relative Motion · Class 11 Notes

Class 11 notes on relative motion: river crossing, rain and the umbrella, and closest approach. One subtraction decides every question in the topic. Free to read or download.

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In short

Relative motion runs on one rule: the velocity of A relative to B is v(A) minus v(B). Get that subtraction the right way round and river crossing, the rain-and-umbrella problem and closest approach all collapse into ordinary kinematics. Getting it backwards is the single most common way marks are lost here.

Contents
  1. ·How to Read This
  2. 1The One Rule and the order that trips people
  3. 2River Crossing shortest time against shortest path
  4. 3Rain and the Umbrella why you tilt it forward
  5. 4Closest Approach two bodies, one frame
  6. 5Relative Motion in a Plane using components
  7. ★Relative Motion · Fact Sheet
0

How to Read This

  • Relative motion is not a new law. It is a change of viewpoint.
  • Pick the right frame and a hard problem becomes a simple one.
  • Blue marks a defining fact. Red marks a sign trap, and almost every error here is a sign.
1

The One Rule

  • The velocity of A as seen by B is v(AB) = v(A) − v(B).
  • The order matters. v(AB) and v(BA) are equal in size and opposite in direction.
  • Read it as: the thing you want, minus the thing you are standing on.
  • In the frame of B, B is at rest by definition. Everything else moves relative to it.
  • The same subtraction works for acceleration: a(AB) = a(A) − a(B).
Relative velocity is vector subtraction, so reverse one arrow and add.ADDING VECTORS: TRIANGLE AND PARALLELOGRAM ARE THE SAME RULEABR = A + Btail to head, and R closes the triangleR = √(A² + B² + 2AB cosθ)tanα = B sinθ / (A + B cosθ)THREE SPECIAL CASESθ = 0: R = A + B, the maximumθ = 180°: R = |A − B|, the minimumθ = 90°: R = √(A² + B²)
Relative velocity is vector subtraction, so reverse one arrow and add.
2

River Crossing

  • A boat of speed v crosses a river of width d, with the water flowing at u.
  • Shortest time: point straight across. Time = d/v, and the river carries you downstream by (u/v)d. The drift does not affect the time.
  • Shortest path: point upstream at an angle so the drift cancels. Then sinθ = u/v, and the crossing takes d/√(v² − u²).
  • If u is greater than v, the shortest path is impossible. The boat cannot beat the current, so it must drift.
Trap alert
The time to cross depends only on the across-component
  • This is the single most useful fact in the topic.
  • The river flows along the bank, so it has no component across the river.
  • So it cannot change the crossing time when you steer straight across.
  • It only changes where you land.
3

Rain and the Umbrella

  • Rain falls with velocity v(rain). You walk with velocity v(man).
  • You experience v(rain) − v(man), which tilts the rain toward you.
  • So you tilt the umbrella forward, into the direction you are walking.
  • tanθ = v(man) / v(rain), measured from the vertical.
  • Walk faster and you must tilt it further forward, which matches everyday experience and is a good check on your answer.
4

Closest Approach

  • Two bodies move with constant velocities. When are they nearest?
  • Sit in the frame of one of them. Now it is at rest and the other moves in a straight line.
  • The closest approach is then simply the perpendicular distance from the stationary body to that straight line.
  • The time to reach it follows from the relative speed along the line of approach.
  • This reduces a two-body problem to a piece of school geometry.
5

Relative Motion in a Plane

  • Split everything into components and handle each direction separately.
  • The x components subtract, and the y components subtract, independently.
  • Two bodies collide if their relative velocity points exactly along the line joining them.
  • If the relative velocity is perpendicular to that line, the separation is momentarily unchanged.
  • For projectiles launched together, relative acceleration is zero, since both have g. So one moves in a straight line relative to the other.

★ Relative Motion · Fact Sheet

Every rule for revision day.

THE ONE RULE

v(AB) = v(A) − v(B)

The thing you want, minus
the thing you stand on.

ORDER MATTERS

v(AB) = −v(BA)

Same size, opposite direction.
Most errors here are signs.

SHORTEST TIME

Point straight across.

t = d/v, drift = (u/v)d
Current does not change t.

SHORTEST PATH

Point upstream: sinθ = u/v

t = d / √(v² − u²)
Impossible if u > v.

RAIN

tanθ = v(man) / v(rain)

Tilt the umbrella FORWARD
Faster walk, greater tilt.

CLOSEST APPROACH

Sit in one body's frame.

Answer is the perpendicular
distance to the straight line.

COLLISION CONDITION

They collide if the relative

velocity points along the line
joining the two bodies.

TWO PROJECTILES

Relative acceleration is ZERO,

since both have g.
So relative motion is straight.

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