Differentiate parametric curves and integrate vector motion

Translate one moving root into a vector

For j=0, r(t)=2eᵗ(cos t,sin t), t≥0. Define eᵣ=(cos t,sin t) and eθ=(−sin t,cos t). These perpendicular unit vectors rotate with t, so their derivatives must also be included

Work the velocity and acceleration

The product rule gives v=2eᵗ(eᵣ+eθ) and a=4eᵗeθ. Hence speed is 2√2eᵗ. At t=0, r=(2,0), v=(2,2), a=(0,4). Speed is a magnitude, not a velocity component; zero x-acceleration at one time does not mean no acceleration

For r(t)=2eᵗ(cos t,sin t), what is vᵧ(0)?

For r(t)=2eᵗ(cos t,sin t), what is the squared speed at t=0?

Complete the motion extension, ungraded

Derive the velocity and acceleration of R eᵃᵗ(cos ωt,sin ωt), with R>0 and real a,ω. Find speed and distance travelled from 0 to T≥0, including a=0. For the collision question, vertex j has position R eᵃᵗ(cos(ωt+2πj/3),sin(ωt+2πj/3)) for j=0,1,2. Explain whether two vertices can collide at the same finite time

Model: retain the rotating basis and the exceptional case

Now eᵣ′=ωeθ and eθ′=−ωeᵣ. Therefore v=R eᵃᵗ(aeᵣ+ωeθ), a⃗=R eᵃᵗ[(a²−ω²)eᵣ+2aωeθ], and speed=R eᵃᵗ√(a²+ω²). Distance is ∫₀ᵀR eᵃᵗ√(a²+ω²) dt. For a≠0, this is R√(a²+ω²)(eᵃᵀ−1)/a; for a=0 it is R|ω|T, including zero for ω=0. The quotient stays nonnegative when a<0. The angular separation between vertices is 2π/3, giving a separation of √3 R eᵃᵗ. This is nonzero at every finite t, so no two collide at the same time

For r(t)=R eᵃᵗ(cos ωt,sin ωt), with R=2, a=1, ω=1 and 0≤t≤T=ln2, the distance travelled is c√2. Find c?

Does seeing two plotted paths cross prove that the moving points collide?

Use technology as a check with a record

A graphing tool can plot x=2eᵗ cos t,y=2eᵗ sin t for 0≤t≤ln2. Record the parameter interval and equal axis scales. At t=0 verify (2,0) and the tangent direction (2,2); at ln2 verify radius 4. Numerical differentiation and a plot support checking, but the product-rule derivation supplies the reasoning

Marking guide for the written extension

A complete human-reviewed response defines the moving basis, differentiates both factors, separates speed from velocity, integrates with the a=0 case, and justifies collision using a nonzero difference