Exam 14: Calculus of Vector-Valued Functions

A particle has acceleration initial velocity , and initial position . Decompose into tangential and normal components.

Find an arc length parametrization for the line .

Particle 1 and Particle 2 are flying through space. At time , the position of Particle 1 is given by and the position of Particle 2 is given by . Do the particles ever collide? If so, where?

Evaluate .

Find the speed at time of the particle that is traveling along the curve .

Find the length of the curve for .

Let and A) Compute and then differentiate the resulting function. B) Find using the cross-product rule for differentiation.

Find the length of the curve for .

A projectile is fired from the ground at an angle of with an initial speed of . How far does the projectile travel in the horizontal direction? Recall that is the acceleration due to gravity on the Earth's surface. Approximate your answer to the nearest tenth of a kilometer.

Find the curvature of at .

The position of a particle is given by , . What is the limit of the particle's speed as

Find the curve traced by the following vector valued function and describe it with a Cartesian equation.

Find the length of the curve described by the vector function .

If a shuttle orbits about 300 km from the surface of Earth and Earth's radius is 6500 km, how long would it take for the shuttle to go around Earth? Approximate your answer to the nearest hundredth of an hour. (The orbital distance is measured from the center of Earth, and the mass of Earth is kg.) Recall .

A particle moves along the curve . Compute the velocity and acceleration vectors at .

Find a parametrization of the osculating circle at if .

A particle moves along the curve . Decompose the acceleration into tangential and normal components.

Find where .

Compute .

The path of a certain particle is parametrized by , . At what time is the minimum speed of the particle achieved?