Nonuniform straight-line motion Consider the motion of an object given by the position function r ( t ) = f ( t ) 〈 a , b , c 〉 + 〈 x 0 , y 0 , z 0 〉 , for t ≥ 0 , where a, b, c , x 0 , y 0 , and z 0 are constants, and f is a differentiable scalar function, for t ≥ 0. a. Explain why this function describes motion along a line. b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?
Nonuniform straight-line motion Consider the motion of an object given by the position function r ( t ) = f ( t ) 〈 a , b , c 〉 + 〈 x 0 , y 0 , z 0 〉 , for t ≥ 0 , where a, b, c , x 0 , y 0 , and z 0 are constants, and f is a differentiable scalar function, for t ≥ 0. a. Explain why this function describes motion along a line. b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?
Solution Summary: The author explains why the function describes motion along a line.
Nonuniform straight-line motion Consider the motion of an object given by the position function r(t) = ƒ(t)⟨a, b, c⟩ + ⟨x0, y0, z0⟩, for t ≥ 0,where a, b, c, x0, y0, and z0 are constants, and ƒ is a differentiable scalar function, for t ≥ 0.a. Explain why r describes motion along a line.b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?
Calculus
Assume that the level surface equation x3+y3+z3+6xyz = 1 defines z implicitly as a function of x and y. Find zx(0, −1) and zy(0, −1). Use that information to find the equation of the plane tangent to the given level surface at the point corresponding to x = 0 and y = −1−1.
Use the model for projectile motion, assuming there is no air resistance.
Find the vector-valued function for the path of a projectile launched at a height of 10 feet above the ground with an initial velocity of 72 feet per second and at an angle of 30° above the horizontal.
r(t) -
Use a graphing utility to graph the path of the projectile.
r(t)
r(t)
20
50
100
150
200
250
300
50
100
150
200
250
300
r(t)
r(t)
50
100
150
200
250
300
50
100
150
200
250
300
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.