Physics
Relative Motion
River Boat Problems: Crossing Directly Opposite
Class 11
⚡ Quick Summary
To cross a river and end up exactly opposite where you started, you need to swim at an angle against the current. The angle and your swimming speed must be just right to cancel out the river's flow. Use vectors! Break down speeds into x and y components.
v<sub>m,g</sub> = v<sub>m,r</sub> + v<sub>r,g</sub>
- Let vm,g be the velocity of the man with respect to the ground (which should be perpendicular to the river flow).
- Let vr,g be the velocity of the river with respect to the ground.
- Let vm,r be the velocity of the man with respect to the river (swimming speed in still water).
- The vector sum of vm,r and vr,g must equal vm,g.
River Boat Problems: Swimming at an Angle
Class 11
⚡ Quick Summary
If you swim at an angle to the river flow, your actual path depends on both your swimming speed and the river's speed. To find out where you'll end up and how long it'll take, break down your swimming speed into components parallel and perpendicular to the river flow.
v<sub>m,g</sub> = v<sub>m,r</sub> + v<sub>r,g</sub>
- Let vr,g = velocity of the river with respect to the ground.
- Let vm,r = velocity of the man with respect to the river.
- Let vm,g = velocity of the man with respect to the ground.
River Boat Problems: Time to cross the river
11
⚡ Quick Summary
When a boat is sailing perpendicular to the river flow, the time to cross depends on the boat's velocity with respect to water and the width of the river. The faster the boat and the narrower the river, the quicker you'll get across!
t = d / v
If a boat is sailing at a velocity *v* with respect to the water, in a direction perpendicular to the river of width *d*, the time taken *t* to cross the river is given by:
* *t* = *d* / *v*
River Boat Problems: Drift
11
⚡ Quick Summary
The 'drift' is how far downstream the boat lands from directly across. This depends on the river's speed and the time it takes to cross. A fast river or a slow boat means a bigger drift!
Drift = (River Speed) * (Time to cross) = v_river * t
The distance from the point directly opposite the starting point where the boat reaches the opposite bank (the 'drift') depends on the velocity of the river current and time taken to cross the river. Drift = (River Speed) x (Time to cross).
Swimmer Crossing a River at an Angle
11
⚡ Quick Summary
If a swimmer heads at an angle to the flow, the crossing time changes! Figuring out the right angle can help minimize how long it takes to cross, or to reach a specific point on the other side.
Time to cross = d / (v * cos(θ)) where d is river width and v is swimmer's speed with respect to water.
When a swimmer heads in a direction making an angle θ with the flow, the component of the swimmer's velocity with respect to water that's perpendicular to the river flow determines the crossing time. The shortest time to cross the river is achieved when the swimmer heads perpendicular to the flow.
Minimum Distance Walked After Crossing River
11
⚡ Quick Summary
Sometimes, you need to land *exactly* opposite your starting point. If the river pushes you off course, you'll have to walk back! The goal is to minimize that walk.
This is a problem solving concept, no specific formula.
To reach the point directly opposite the starting point, the swimmer must swim at an angle such that the component of their velocity along the river cancels out the river's velocity. If they land somewhere else, they must walk the difference to reach the target. This problem often involves vector components.
Aeroplane and Wind Velocity
11
⚡ Quick Summary
Planes have to fight the wind! The plane's 'airspeed' (how fast it moves through the air) and the wind's speed and direction combine to determine the plane's actual path over the ground.
v_ground = v_air + v_wind (Vector addition)
The resultant velocity of the aeroplane is the vector sum of the velocity of the aeroplane with respect to the air and the velocity of the wind. This resultant velocity determines the actual direction and speed of the aeroplane relative to the ground.
Sound and Wind Velocity
11
⚡ Quick Summary
Wind affects how fast sound travels! It's like a river for sound waves. If the wind blows towards you, the sound arrives sooner.
v + u = x/t1, v - u = x/t2; Solving these equations yields v and u.
The velocity of sound in still air *v* and the velocity of wind *u* can be determined by measuring the time taken for sound to travel between two points when the wind is blowing from one point to the other and vice versa. The effective speed of sound is the sum or difference of the speed of sound in still air and the wind speed, depending on the direction.
Particles Moving Towards Each Other
11
⚡ Quick Summary
Imagine particles chasing each other in a circle! They each move towards the next one, creating a spiral until they meet. It's all about figuring out how their relative speeds affect the meeting time.
t = a / (v - v*cos(θ)) where a is side length and v is speed. For a hexagon θ = 60 degrees.
When particles are situated at the corners of a regular polygon and each moves towards the next particle, they will eventually meet at the center. The time taken depends on the initial separation, the speed of the particles, and the geometry of the polygon. The component of one particle's velocity towards the meeting point can be used to find the time.