Number and Algebra
Functions
Geometry & Trigonometry
Statistics & Probability
Calculus
Find kkk such that the trajectory r(t)=⟨kt, t2, 3⟩\mathbf r(t)=\langle k t,\,t^2,\,3\rangler(t)=⟨kt,t2,3⟩ passes through the point (8,4,3)(8,4,3)(8,4,3) when t=2t=2t=2.
A particle moves according to r(t)=⟨3,−2,5⟩+t⟨2,1,−4⟩\mathbf r(t)=\langle3,-2,5\rangle+t\langle2,1,-4\rangler(t)=⟨3,−2,5⟩+t⟨2,1,−4⟩. Find its constant speed.
The position of a particle is r(t)=⟨t2, 3t, 4⟩\mathbf r(t)=\langle t^2,\,3t,\,4\rangler(t)=⟨t2,3t,4⟩. Find the velocity vector and the speed at t=2t=2t=2.
The path of a particle is r(t)=⟨t, t2, t3⟩\mathbf r(t)=\langle t,\,t^2,\,t^3\rangler(t)=⟨t,t2,t3⟩. Find its acceleration vector at t=2t=2t=2.
A particle has constant acceleration a=⟨2,−1,3⟩\mathbf a=\langle2,-1,3\ranglea=⟨2,−1,3⟩ and initial velocity v(0)=⟨1,0,0⟩\mathbf v(0)=\langle1,0,0\ranglev(0)=⟨1,0,0⟩. Find v(t)\mathbf v(t)v(t) and its speed at t=4t=4t=4.
A particle moves on the line r(t)=⟨1,2,3⟩+t⟨4,−2,1⟩\mathbf r(t)=\langle1,2,3\rangle+t\langle4,-2,1\rangler(t)=⟨1,2,3⟩+t⟨4,−2,1⟩. At what time is the particle 10 units from the origin?
Find the value of kkk such that the object with position vector r=⟨2,1,4⟩+k⟨5,9,1⟩\mathbf r=\langle2,1,4\rangle+k\langle5,9,1\rangler=⟨2,1,4⟩+k⟨5,9,1⟩ has speed 500500500.
A particle has velocity v(t)=⟨6t,4,−2t⟩\mathbf v(t)=\langle6t,4,-2t\ranglev(t)=⟨6t,4,−2t⟩ and initial position r(0)=⟨1,2,3⟩\mathbf r(0)=\langle1,2,3\rangler(0)=⟨1,2,3⟩. Find the position vector r(t)\mathbf r(t)r(t).
A particle moves with r(t)=⟨5cost, 5sint, 3t⟩\mathbf r(t)=\langle5\cos t,\,5\sin t,\,3t\rangler(t)=⟨5cost,5sint,3t⟩. Determine its speed as a function of ttt.
A particle moves according to r(t)=⟨et,t,sint⟩\mathbf r(t)=\langle e^t,t,\sin t\rangler(t)=⟨et,t,sint⟩. Find its instantaneous speed at t=0t=0t=0.
A projectile is launched from the origin with initial velocity ⟨10,10,0⟩\langle10,10,0\rangle⟨10,10,0⟩ and acceleration ⟨0,0,−9.8⟩\langle0,0,-9.8\rangle⟨0,0,−9.8⟩. Its position is given by
Find its speed at t=1t=1t=1.
The path of a particle is r(s)=⟨s,s3,s2⟩\mathbf r(s)=\langle s, s^3, s^2\rangler(s)=⟨s,s3,s2⟩, where sss denotes time. Find all values of sss for which the speed is 101010.
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