Section Summary

Kinematics in Two Dimensions: An Introduction

• The shortest path between any two points is a straight line. In two dimensions, this path can be represented by a vector with horizontal and vertical components.
• The horizontal and vertical components of a vector are independent of one another. Motion in the horizontal direction does not affect motion in the vertical direction, and vice versa.

Vector Addition and Subtraction: Graphical Methods

• The graphical method of adding vectors A𝐴 and B𝐵 involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector R𝑅 is defined such that A+B=RA+B=R. The magnitude and direction of R𝑅 are then determined with a ruler and protractor, respectively.
• The graphical method of subtracting vector B𝐵 from A𝐴 involves adding the opposite of vector B𝐵, which is defined as −B−𝐵. In this case, A–B=A+(–B)=RA–B=A+(–B)=R. Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector R𝑅.
• Addition of vectors is commutative such that A+B=B+AA+B=B+A .
• The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
• If a vector A𝐴 is multiplied by a scalar quantity c𝑐, the magnitude of the product is given by cAcA. If c𝑐 is positive, the direction of the product points in the same direction as A𝐴; if c𝑐 is negative, the direction of the product points in the opposite direction as A𝐴.

Vector Addition and Subtraction: Analytical Methods

• The analytical method of vector addition and subtraction involves using the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector.
• The steps to add vectors A𝐴 and B𝐵 using the analytical method are as follows:Step 1: Determine the coordinate system for the vectors. Then, determine the horizontal and vertical components of each vector using the equationsAxBx==AcosθBcosθ𝐴𝑥=𝐴cos𝜃𝐵𝑥=𝐵cos𝜃andAyBy==AsinθBsinθ.𝐴𝑦=𝐴sin𝜃𝐵𝑦=𝐵sin𝜃.Step 2: Add the horizontal and vertical components of each vector to determine the components Rx𝑅𝑥 and Ry𝑅𝑦 of the resultant vector, RR:Rx=Ax+Bx𝑅𝑥=𝐴𝑥+𝐵𝑥andRy=Ay+By.𝑅𝑦=𝐴𝑦+𝐵𝑦.Step 3: Use the Pythagorean theorem to determine the magnitude, R𝑅, of the resultant vector RR:R=R2x+R2y−−−−−−−√.𝑅=𝑅𝑥2+𝑅𝑦2.Step 4: Use a trigonometric identity to determine the direction, θ𝜃, of RR:θ=tan−1(Ry/Rx).𝜃=tan−1(𝑅𝑦/𝑅𝑥).

Projectile Motion

• Projectile motion is the motion of an object through the air that is subject only to the acceleration of gravity.
• To solve projectile motion problems, perform the following steps:
  1. Determine a coordinate system. Then, resolve the position and/or velocity of the object in the horizontal and vertical components. The components of position s𝑠 are given by the quantities x𝑥 and y𝑦, and the components of the velocity v𝑣 are given by vx=vcosθ𝑣𝑥=𝑣cos𝜃 and vy=vsinθ𝑣𝑦=𝑣sin𝜃, where v𝑣 is the magnitude of the velocity and θ𝜃 is its direction.
  2. Analyze the motion of the projectile in the horizontal direction using the following equations:Horizontal motion(ax=0)Horizontal motion(𝑎𝑥=0)x=x0+vxt𝑥=𝑥0+𝑣𝑥𝑡vx=v0x=vx=velocity is a constant.𝑣𝑥=𝑣0𝑥=vx=velocity is a constant.
  3. Analyze the motion of the projectile in the vertical direction using the following equations:Vertical Motion(assuming positive is upay=−g=−9.80m/s2)Vertical Motion(assuming positive is up𝑎𝑦=−𝑔=−9.80m/s2)y=y0+12(v0y+vy)t𝑦=𝑦0+12(𝑣0𝑦+𝑣𝑦)𝑡vy=v0y−gt𝑣𝑦=𝑣0𝑦−gty=y0+v0yt−12gt2𝑦=𝑦0+𝑣0𝑦𝑡−12gt2v2y=v20y−2g(y−y0).𝑣𝑦2=𝑣0𝑦2−2𝑔(𝑦−𝑦0).
  4. Recombine the horizontal and vertical components of location and/or velocity using the following equations:s=x2+y2−−−−−−√𝑠=𝑥2+𝑦2θ=tan−1(y/x)𝜃=tan−1(𝑦/𝑥)v=v2x+v2y−−−−−−√𝑣=𝑣𝑥2+𝑣𝑦2θv=tan−1(vy/vx).𝜃v=tan−1(𝑣𝑦/𝑣𝑥).
• The maximum height hℎ of a projectile launched with initial vertical velocity v0y𝑣0𝑦 is given byh=v20y2g.ℎ=𝑣0𝑦22𝑔.
• The maximum horizontal distance traveled by a projectile is called the range. The range R𝑅 of a projectile on level ground launched at an angle θ0𝜃0 above the horizontal with initial speed v0𝑣0 is given byR=v20sin2θ0g.𝑅=𝑣02sin2𝜃0𝑔.

Addition of Velocities

• Velocities in two dimensions are added using the same analytical vector techniques, which are rewritten asvx=vcosθ𝑣𝑥=𝑣cos𝜃vy=vsinθ𝑣𝑦=𝑣sin𝜃v=v2x+v2y−−−−−−√𝑣=𝑣𝑥2+𝑣𝑦2θ=tan−1(vy/vx).𝜃=tan−1(𝑣𝑦/𝑣𝑥).
• Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies dramatically with reference frame.
• Relativity is the study of how different observers measure the same phenomenon, particularly when the observers move relative to one another. Classical relativity is limited to situations where speed is less than about 1% of the speed of light (3000 km/s).