The position vector of an object

Question

The position vector of an object r(t) = < t^2/2 cos sqrt(2t)i + t^2/ sqrt(2) sin sqrt(2t)j + 22-tk. Calculate the time t takes for the object to reach z = 0. Then compute the distance L traveled by the object from time t = 8 until t = 19. What is its speed |v| at t = 8 seconds?

Answer 1

a) Calculate the time t takes for the object to reach z = 0.

t is a just a parameter, but in this problem it takes the meaning of time. To find the time when the z-component reaches zero we can just solve the equation.

z(t) = 22 - t = 0 ⇒ t = 22.

b) Then compute the distance L traveled by the object from time t = 8 until t = 19.

For this part we can just use the equation for distance of a curve. These integrals can get quite tough so we might need to solve it numerically.

l = ∫_a^b√(dx/dt)² + (dy/dt)² + (dz/dt)² dt (length equation for parametric curve)

x = t²/√2 cos(√2t), dx/dt = √2 t cos(√2t) - 1/2 t^3/2 sin(√2t)

y = t²/√2 sin(√2t), dy/dt = √2 t sin(√2t) + 1/2 t^3/2 cos(√2t)

z = 22- t, dz/dt = -1

Notice that (dx/dt)²+(dy/dt)² + (dz/dt) = 1/4(t³ + 8t² + 4)

l = 1/2 ∫₈¹⁹√t³ + 8t² + 1 dt

There is not a simple antiderivative for this so using a calculator or algebra website should be fine. I used WolframAlpha.

l = 349.53

c) What is its speed |v| at t = 8 seconds?

For this we can just use the speed equation. Notice this is the magnitude of the velocity.

ΙvΙ = √(dx/dt)² + (dy/dt)² + (dz/dt)²

ΙvΙ = 1/2 √t³ + 8t² + 4 = 1/2 √257

Please double check this for algebra mistakes.