A.1 · 9e · A.1 Kinematics · relative velocity, vector addition, components

Adding velocities: crosswind landing and the umbrella

Land a plane in a crosswind, then keep dry in the rain. Two games, one triangle of velocities.

SL+HLHarder28 minFree

Hook

To fly straight down a runway in a crosswind, the pilot does not point the nose along the runway. To stay dry in the rain, you do not hold the umbrella straight up. Different scenes, same triangle.

The rule

1 / 7

Velocities add as vectors

Top view of the same landing, flown two ways. The black arrow is the sum of the other two: it is where the aircraft actually goes.

v_{over the ground} = v_{through the air} + v_{of the air (wind)}

draw them head to tail
the sum is the third side of the triangle
it is the velocity you actually have over the ground

VectorIn the crosswind landing

through the air200 km h⁻¹ along the nosewhat the pilot controls

of the air50 km h⁻¹ sidewaysthe wind

over the groundtheir vector sumwhere the plane really goes

Check 1 of 6

A plane flies at 200 km h⁻¹ through the air in a 50 km h⁻¹ crosswind. To fly straight along the runway, the nose must point…

Check 2 of 6

For that landing, the plane’s speed along the runway is about…

Check 3 of 6

Rain falls vertically at 6.0 m s⁻¹. You run at 3.0 m s⁻¹. From you, the rain seems to come at what angle from the vertical?

Check 4 of 6

For the same runner and rain, the speed of the rain relative to the runner is…

Check 5 of 6

You walk faster in the same rain. To stay dry, the umbrella should lean…

Check 6 of 6

A pilot ignores the wind and points the nose straight along the runway. The plane will…

Apply

A swimmer can swim at 1.5 m s⁻¹ in still water and wants to cross a river straight across to the opposite bank. The river flows at 0.60 m s⁻¹. At what angle upstream should she aim, and how fast does she cross?

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Remember earlier lessons

From lesson A.1 · 9c

The landing speed of a parcel is found from its components by…

From lesson A.1 · 9d

A ball thrown at 30° at 20 m s⁻¹ has u_y equal to…

Summary card

Key points

• Velocities add as vectors, head to tail: v(ground) = v(air) + v(wind).
• Crosswind landing: point the nose into the wind, sinθ = W / V, ground speed √(V² − W²).
• Rain on a walker: v(rain relative) = v(rain) − v(walker), tanθ = W / v, lean the umbrella forward.
• Sketch the triangle, work out which sides you know, then choose sin, cos, tan or Pythagoras.

Formulas

v_ground = v_air + v_wind
sinθ = W / V
tanθ = W / v
v_rel = v_A − v_B

Key terms

Relative velocity: the velocity of one thing as seen by another: v(A) − v(B)
Crab angle: the angle a pilot points the nose into the wind so the plane tracks straight

Common errors

• Pointing the nose at the destination in a crosswind.
• Adding the speeds instead of using Pythagoras.
• Forgetting that the walker sees the rain slanted.

Checks solved: 0 of 6

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