* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Lecture 4
Survey
Document related concepts
Jerk (physics) wikipedia , lookup
Specific impulse wikipedia , lookup
Fictitious force wikipedia , lookup
Four-vector wikipedia , lookup
Bra–ket notation wikipedia , lookup
Faster-than-light wikipedia , lookup
Classical mechanics wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Relativistic angular momentum wikipedia , lookup
Derivations of the Lorentz transformations wikipedia , lookup
Equations of motion wikipedia , lookup
Classical central-force problem wikipedia , lookup
Renormalization group wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Velocity-addition formula wikipedia , lookup
Transcript
Physics I 95.141 LECTURE 4 9/15/10 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Exam Prep Problem • Vince Carter’s vertical leap is 43”. – A) (10pts) With what initial vertical velocity does Carter leave the ground? – B) (10pts) What is his hang time? – C) (10pts) Assuming Carter leaps straight up at t=0s and lands at just after t=T, draw the vectors for James’ displacement, velocity, and acceleration at: • • • • • i) The instant he leaves the ground (t=0s) ii) t=T/4 iii) t=T/2 iv) t=3T/4 v) t=T 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Exam Prep Problem • Vince Carter’s vertical leap is 43”. – A) (10pts) With what initial vertical velocity does James leave the ground? 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Exam Prep Problem • Vince Carter’s vertical leap is 43”. – B) (10pts) What is his hang time? 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Exam Prep Problem • Vince Carter’s vertical leap is 43”. – C) (10pts) Assuming Carter leaps straight up at t=0s and lands at just after t=T, draw the vectors for James’ displacement, velocity, and acceleration at: t 0 y v a t T4 y v a t T2 y v a 95.141, F2010, Lecture 4 Department of Physics and Applied Physics t 3T 4 y v a t T y v a Outline • Lecture 3 Review • Vector Kinematics • Relative Motion • What do we know? – – – – Units/Dimensions/Measurement/SigFigs Kinematic equations Freely falling objects Vectors 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Lecture 3 Review • Freely falling body problems – Batman’s bat-hook • Scalars and Vectors • Graphical description of vectors and vector addition. • Vector components • Mathematical description of vector addition (addition of components) • Unit Vectors 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Vector Kinematics • We can now do kinematics in more than one dimension – This is helpful, because we live in a 3D world! • We previously described displacement as Δx, but this was for 1D, where motion could only be positive or negative. • In more than 1 dimension, displacement is a vector v r 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Vector Kinematics x x2 x1 y r r2 r1 x Now, instead of describing displacement in terms of either vertical or horizontal position, we can talk about a displacement vector! 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Vector Kinematics • In unit vectors, we can write the displacement vector as: • We can now rewrite our expression for average velocity: 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Vector Kinematics • Average velocity only tells part of the story • Just like for motion in 1D, we can let Δt get smaller and smaller…. • Gives instantaneous velocity vector: lim r v t 0 t 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Velocity Vector • The magnitude of the average velocity vector is NOT equal to the average speed. • But the magnitude of the instantaneous velocity vector is equal to the instantaneous speed at that time 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Instantaneous Velocity (math) • To find the instantaneous velocity, we can take the derivative of the position vector with respect to time: r x (t )iˆ y (t ) ˆj z (t )kˆ 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Example • Say we are given the position of an object to be: r (t ) (4t 2 1)iˆ 2e t ˆj sin( 2t )kˆ • Can we find the velocity as a function of time? 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Acceleration Vector • Average acceleration: • Instantaneous acceleration 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Example Problem • Imagine we are given the position of an object as a function of time – Find displacement at t=1s and t=3s – Find velocity and acceleration as a function of time – Find velocity and acceleration at t=3s r (t ) 4(m s )t 2(m s 2 )t 2 iˆ 3(m) 3(m s 3 )t 3 ˆj 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Example Problem • Imagine we are given the position of an object as a function of time – Find displacement at t=1s and t=3s – Find velocity and acceleration as a function of time – Find velocity and acceleration at t=3s r (t ) 4(m s )t 2(m s 2 )t 2 iˆ 3(m) 3(m s 3 )t 3 ˆj 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Example Problem • Imagine we are given the position of an object as a function of time – Find displacement at t=1s and t=3s – Find velocity and acceleration as a function of time – Find velocity and acceleration at t=3s 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Example Problem • Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0). – A) Give the initial velocity vector of the object – B) Plot x(t) vs. t – C) Plot y(t) vs. t – D) Plot the object’s trajectory in the xy plane 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Example Problem • Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0).. 20 vy – A) Give the initial velocity vector of the object vx 20 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Example Problem • Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0). – Before we solve B-D, let’s determine equations of motion (METHOD I) 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Example Problem • Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0). – Before we solve B-D, let’s determine equations of motion (METHOD II) 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Example Problem • Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0). x (t ) vox t 10t – B) Plot x(t) vs t x(t) 100 t(s) 10 95.141, F2010, Lecture 4 Department of Physics and Applied Physics time x(t) 0 0m 1 10m 2 20m 5 50m 10 100m Example Problem • Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, starts at (0,0). – C) Plot y(t) vs t 1 2 y(m) y (t ) v oy t at 15t 2.5t 2 2 t(s) 95.141, F2010, Lecture 4 Department of Physics and Applied Physics time y(t) 0 0m 1 12.5m 2 20m 3 22.5m 4 20m 5 12.5 10 -100 x (t ) 10t Example Problem y(t ) 15t 2.5t 2 D) Plot object trajectory • Choose coordinate system y( x) 1.5 x .025 x 2 y(m) x(m) 95.141, F2010, Lecture 4 Department of Physics and Applied Physics time x(t) y(t) 0 0m 0m 1 10m 12.5m 2 20m 20m 3 30m 22.5m 4 40m 20m 5 50 12.5 10 100 -100 Relative Velocity • So far we have looked at adding displacement vectors • May also find situations where we need to add velocity vectors 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Relative Velocity • Two velocities: 5m/s 25m/s 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Relative Velocity • In this case, our hero would presumably prefer not to be decapitated by the bridge • So we are interested in his velocity relative to the bridge • He is on a train moving at +25 m/s relative to the bridge • His velocity relative to the train is -5m/s • So his velocity relative to the bridge is: 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Relative Velocity • What is Kirk’s velocity when he hits the ground? – Assume he leaps when car is moving 20m/s 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Relative Velocity (Example 1) • Imagine you are on a barge floating down the river with the current vriver 3 m s i • You walk diagonally across the barge with a velocity v walk on barge 2 m s iˆ 2 m s ˆj • What is your velocity with respect to the water? • With respect to the river bank? 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Another River Problem • A boat’s speed in still water is 1.85m/s. If you want to directly cross a stream with a current 1.2m/s, what upstream angle should you take? 95.141, F2010, Lecture 4 Department of Physics and Applied Physics Today We Learned…. • Vector kinematics – – – – – – Displacement vector Average velocity vector Inst. Velocity vector Average acceleration vector Inst. Acceleration vector Vector equations of motion • Relative Velocity 95.141, F2010, Lecture 4 Department of Physics and Applied Physics