Lessons 1–2 · 2 lessons · N. Sh. Turdiyev, K. A. Tursunmetov, A. G. Ganiyev, K. T. Suyarov, J. E. Usarov, A. K. Avliyoqulov. Physics Grade 10, 1st edition. Niso Poligraf Publishing House, Tashkent, 2017
2
Types of mechanical motion. The principle of independence of motions
Textbook: pp. 5–7
GoalRecall the types of mechanical motion and apply the principle of independence of motions and the addition of velocities.
New words
trajectory · trayektoriyaprinciple of independence · mustaqillik prinsipiresultant velocity · natijaviy tezlikprojection · proyeksiya
Explanation
By trajectory a motion is straight or curved, and by speed it is uniform or non-uniform. In uniform straight-line motion s = υt; in uniformly variable motion s = υ₀t + at²/2 and υ = υ₀ + at (a > 0 if speeding up, a < 0 if slowing down). Uniform motion along a circle is described by the linear speed υ, angular speed ω, period T and frequency ν. A body often takes part in several motions at once (a ship on a river, a person walking inside a moving train). Experiment shows that these motions do not depend on each other; this is the principle of independence of motions. Therefore the resultant velocity is the vector sum of the velocities: υ = υ₁ + υ₂ + … + υₙ, and in coordinates each axis is written separately: x = x₀ + υₓt + aₓt²/2, y = y₀ + υᵧt + aᵧt²/2. Velocities in opposite directions are subtracted; mutually perpendicular velocities are added by the Pythagorean theorem: υ = √(υ₁² + υ₂²).
Worked examples
A boat moves at 18 km/h in still water and the current is 2 km/h. Time for 40 km downstream: t = s/(υ₁ + υ₂) = 40/(18 + 2) = 2 h; upstream: t = 40/(18 – 2) = 2.5 h.
A boat moves at 4 m/s relative to the water, perpendicular to the bank; the current is 3 m/s and the river is 100 m wide. Resultant speed υ = √(4² + 3²) = 5 m/s. The crossing time depends only on the perpendicular motion: t = 100/4 = 25 s. In that time the current carries the boat 3·25 = 75 m downstream.
Class activity
With the teacher: one student walks slowly along the room while another rolls a ball past them on a desk. Count steps and compare the ball’s speed relative to the floor and relative to the walker. Do not run or jump off desks.
Practice
1
State the principle of independence of motions.
The motions a body takes part in do not depend on each other: each goes on as if the others were absent.
2
A train moves at 20 m/s. A passenger inside walks in the direction of travel at 1 m/s. What is the speed relative to the ground in m/s?
21
3
A river flows at 2 m/s. A motor boat moves upstream at 7 m/s relative to the water. What is its speed relative to the bank in m/s?
5
4
Why does a swimmer who swims perpendicular to the bank reach the opposite bank in the same time whatever the current speed?
The perpendicular motion does not depend on the current; the time is set only by the width and the speed relative to the water; the current only carries the swimmer downstream.