One-dimensional kinematics

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Claribel Gibson
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1 Phsics 45 Formula Sheet Eam One-dimensional kinematics Vectors displacement: Δ f i total distance traveled average speed total time Δ f i average velocit: vav t f ti Δ instantaneous velocit: v lim Δ t v f average acceleration: aav t f d vi t dv d instantaneous acceleration: a lim One-dimensional motion with constant acceleration: () v v + at () + ( v + v)t (3) + v t + at v v + a (4) ( ) Free fall (positive direction for taken to be upward) and a g in the above 4 equations of kinematics: () v v gt () + ( v + v)t (3) + v t gt v v g (4) ( ) i Vectors in -D If a vector A is written in component form as A A ˆ + A ˆ A, A then: Getting magnitude and direction of A from the components: A A + A (magnitude of A ) A θ arctan (direction of A ) A

2 Formula Sheet Eam Page Getting components from magnitude and direction: A A cosθ ( component of A ) A sinθ ( component of A θ understood to be the angle that ) } A makes with the positive ais A Vectors in 3-D If A A ˆ + A ˆ + A ˆ A, A, A then: A A + A + A -D Kinematics position vector: r () t () t ˆ + ( t) ˆ ( t), ( t) Δr Δ Δ average velocit: v av, dr d d instantaneous velocit: v, average acceleration: aav, instantaneous acceleration: dv dv dv a, d r d d a, instantaneous speed v (magnitude of the instantaneous velocit) 3-D kinematics position vector: r () t () t ˆ + ( t) ˆ + ( t) ˆ ( t), ( t), ( t) Δr Δ Δ Δ average velocit: v av,, dr d d d instantaneous velocit: v,, average acceleration: aav,, instantaneous acceleration: dv dv dv dv a,, d r d d d a,, instantaneous speed v (magnitude of the instantaneous velocit)

3 Formula Sheet Eam Page 3 Projectile Motion direction (motion with constant velocit): a v v v t + direction (free fall... positive direction for taken to be upward): () v v gt () + ( v + v )t (3) + v t gt v v g Relative Motion (4) ( ) vpa vpb + vba Newton s Laws of Motion Two broad categories of forces: contact forces (objects in contact with one another) and field forces (objects not in contact with one another). Gravit is the onl field force we will deal with in this course. Weight: w mg st law: v is constant unless object (or sstem) eperiences net eternal force. nd law : F ma implies three statements, in general: F ma, F ma, and F ma *for sstems of objects, F et mssass 3 rd law: Whenever one object eerts a force on a second object, the second eerts a force on the first; these two forces are equal in magnitude and opposite in direction: F F Equilibrium An object is said to be in equilibrium (reall in translational equilibrium) if: F a *reall implies three separate requirements for translational equilibrium: ) F a ) F a 3) F a Friction forces fs μs fk μk n n

4 Formula Sheet Eam Page 4 Circular Motion nd law (centripetal direction): ( F ) marad v Radial (centripetal) acceleration: a rad r If there is a tangential acceleration a t also, then: rad a t dv Dot Product (Scalar Product) For an two vectors A A, A, A A B AB cosθ, in which: and B B, B, B : A A + A + A Work B B + B + B Alternative (equivalent) definition of dot product: A B A B + A B + A B Work Variable Forces: W F dr (general definition of work) If F has onl an component and this component depends on, then: f W F( ) d i Constant Forces: W F Δ r (-D or 3-D path) W FΔ (-D path) Springs Hooke s law: F k. k the spring constant or the force constant. Potential energ stored in a spring (elastic potential energ): Work-Energ Theorem ΔK W net Uel k Kinetic energ: K mv Power Instantaneous Power:

5 Formula Sheet Eam Page 5 de General Definition: Rate at which energ being supplied b or to a sstem: P If energ comes from work being done, then the power is the rate at which work is done: dw P P F v (alternative definition) Average Power: ΔE ΔW Pav Conservative Forces, Potential Energ, and Conservation of Energ Conservative Forces: Work done b a conservative force: W c ΔU, for some potential energ U Work done is independent of path. Work done going once around closed path is ero. Potential Energ: Gravitational potential energ (near surface of Earth): U grav mg Elastic Potential Energ: Uel Total mechanical energ: E K + U Nonconservative Forces: Wnc Δ E If W nc, E conserved. Force and Potential Energ F( ) ( ) du d Linear Momentum Linear momentum: k p mv p mv p mv p mv dp Δp Fnet Fnet av Δ t Newton s second law: ( ) Impulse net F is net (eternal) force ( Fnet F ) Varing force: t f J F ti Constant force: J F Impulse-momentum theorem: Jnet Δp, in which: t ti (general definition of impulse) f Jnet Fnet (varing force) or Jnet Fnet (constant force)

6 Formula Sheet Eam Page 6 Collisions Two broad categories: head-on and glancing For each categor, three classes:. elastic: p and K conserved. inelastic: p conserved, K not 3. completel inelastic: p conserved, K not, objects stick together Head-on collisions:. elastic: v + m v m v m v (p-conservation) m i i f + f v i v f vi + v f + ( other eq derived in class). inelastic: m vi + mvi mv f + mv f (p-conservation) 3. completel inelastic: v + m v m m v (p-conservation) Glancing (-D) collisions: p p i f p i p f i i ( ) f m + Center of Mass and Sstems of Particles m + m+ + mnn m + m+ + mnn X CM m+ m + + mn Mtot m + m + + mnn m + m + + mnn YCM m+ m + + mn Mtot nd law for sstem: dp ( F ), in which P mv + mv+ + mnvn is total momentum of sstem. et P MtotvCM If mass of sstem is constant, then: ( F ) MtotaCM et

Physics 1 Second Midterm Exam (AM) 2/25/2010

Physics 1 Second Midterm Exam (AM) 2/25/2010 Physics Second Midterm Eam (AM) /5/00. (This problem is worth 40 points.) A roller coaster car of m travels around a vertical loop of radius R. There is no friction and no air resistance. At the top of