Formula sheet: Motion in a Plane

Motion in a Plane14 min readLesson 1 of 7

1.1 Vectors and their algebra A scalar carries only a magnitude. A vector also carries a direction, and it adds by the triangle law. Vectors are written in bold or with an arrow, and the magnitude is written in plain type. Two vectors are equal only when their magnitudes and directions both match, whatever the starting point of the drawing. Addition follows the triangle or parallelogram law. Subtraction is addition of the negative vector. Multiplying a vector by a positive scalar changes only its magnitude, while a negative scalar reverses its direction as well. Resolution breaks a vector into components along chosen perpendicular axes, and those components can be added as scalars.

R = A + B, R = A^{2} + B^{2} + 2 A B cos θ

Resultant of two vectors at an angle theta, with the triangle law

A = A_{x} i^+ A_{y} j^+ A_{z} k^, A = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}

A vector in components and its magnitude

A_{x} = A cos θ, A_{y} = A sin θ, tan θ = \frac{A _{y}}{A _{x}}

Resolution of a vector in a plane, with theta measured from the x-axis

1.2 Motion in a plane Position, displacement, velocity and acceleration are vectors in plane motion. The average velocity divides the displacement vector by the time interval, and the instantaneous velocity is the derivative of the position vector. Acceleration is the derivative of velocity, and the kinematic equations apply component by component when the acceleration is constant. The x and y motions are independent of each other and share only the time.

v = \frac{d r}{d t}, a = \frac{d v}{d t}, r = r_{0} + v_{0} t + \frac{1}{2} a t^{2}

Vector forms of velocity, acceleration and displacement at constant acceleration

1.3 Projectile motion A projectile is a body thrown with an initial velocity at some angle to the horizontal and then moving under gravity alone. Its horizontal velocity stays constant and its vertical motion is uniformly accelerated, so the path is a parabola. The range is largest for a launch angle of 45 degrees, and two complementary angles give the same range. The time of flight, the maximum height and the horizontal range follow from the components of the initial velocity.

T = \frac{2 u sin θ}{g}, H = \frac{u ^{2} sin ^{2} θ}{2 g}, R = \frac{u ^{2} sin 2 θ}{g}

Time of flight, maximum height and horizontal range of a projectile on level ground

y = x tan θ - \frac{g x ^{2}}{2 u ^{2} cos ^{2} θ}

Equation of the parabolic path, taking the launch point as the origin

1.4 Relative velocity and circular motion Relative velocity measures one motion from the frame of another body, and in two dimensions the vectors are subtracted as vectors rather than as lengths. A boat crossing a river and a cyclist riding in rain are the standard cases: the actual velocity is the vector sum of the body's own velocity and the velocity of the medium or frame it moves in. In uniform circular motion the speed is constant but the velocity changes direction continuously, so the acceleration is not zero. That acceleration points toward the centre, is perpendicular to the velocity, and has magnitude v squared over r, which can also be written with the angular speed. The period is the time for one complete revolution.

a_{c} = \frac{v ^{2}}{r} = ω^{2} r, v = ω r, T = \frac{2 π r}{v} = \frac{2 π}{ω}

Centripetal acceleration, the speed and angular speed relation, and the period

Exam

For a projectile, write the two component columns, horizontal and vertical, before touching a formula. Almost every projectile numerical is solved by the vertical motion giving the time and the horizontal motion giving the distance.

in hinglish

Vectors mein magnitude ke saath direction bhi hoti hai, aur addition triangle ya parallelogram law se hota hai. Resolution se vector ko perpendicular components mein todte hain, phir components seedha add hote hain. Motion in a plane mein position, velocity aur acceleration sab vectors hain, aur x tatha y motion sirf time se jude hote hain. Projectile mein horizontal velocity constant rehti hai aur vertical motion g se accelerated hoti hai, isliye path parabola banta hai. Time of flight 2u sin theta / g, height u^2 sin^2 theta / 2g aur range u^2 sin 2 theta / g yaad rakho; 45 degree par range maximum hoti hai. Relative velocity do vectors ka difference hai, isliye direction ke saath subtract karo. Uniform circular motion mein speed constant par direction badalti rehti hai, isliye acceleration centre ki taraf v^2/r hoti hai. Numerical mein pehle components likho, phir equation chuno.