Lecture 4 - Ch 4 notes 3d-motion: general equations of...

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Ch 4 notes 3d-motion: general equations of motion Projectile motion (special case)
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3d-position: 1 r G 2 r G x y r Δ G 11 1 1 22 2 2 ˆ ˆˆ ˆ ˆ ˆ () rx i y j z k i y j z k i y j z k rt xti yt j ztk =+ + + Δ= Δ + Δ + Δ + G G G G 21 or rr r rrr Δ GG G GGG ˆ iy j z k = ++ G
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Examples Plot trajectories: y(x) ˆˆ () (1 2) i (2 )j rt t t =+ + G 2 () () i (5 t t t G
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3d-velocity: ˆˆˆ ij k x yz vv v v =++ G 1 r G 2 r G x y r Δ G ˆ ˆˆ ˆ ˆ avg xy z rx y z vi j k tt t t dr v dt dx dy dz k dt dt dt vi v j vk ΔΔ Δ Δ == + + Δ Δ = =+ + + G G G G ˆ i yj zk + G
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Vector of velocity is tangent to trajectory r G x y ˆˆ ˆ ˆ From where: tan xy y x dx dy vi j v i v j dt dt dy v dy dt dx vd x dt θ =+ = + == = G θ
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Example Find vector of velocity at t=1 s • magnitude • direction Find magnitude of average velocity between t=0 and 2 s 2 ˆˆ ( ) (1 )i (2 5 )j rt t t t =+ + + G
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Difference between avg. and instantaneous velocity
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Example
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Example (train) solution
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3d-acceleration: 22 2 2 ˆˆ ˆ ˆ ˆ y x z xy z dv dv dv dv ai j k a i a j a k dt dt dt dt dx dy dz ij k dt
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Lecture 4 - Ch 4 notes 3d-motion: general equations of...

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