The position vector r describes the path of an object moving in the xy-plane or

Question

The position vector r describes the path of an object moving in the xy-plane or in space. Find the velocity, speed, and acceleration of the object. (a) r(t) = t^i + tj (b) r(t) = 4t, 3 cos t, 3 sin t 2. Use iho given acceleration function to find the velocity and position vectors. Then find the position at time t = 2. (a) a(t) = 2i 4- 3k, v(0) = 4j, r(0) = 0 (b) a(t) = -cos ti - sin tk. v(0) = j + k, r(0) = i 3. A baseball, hit 3 feet above the ground, leaves the bat at an angle of 45^0degree and is caught by an outfielder 3 feet above the ground and 300 feet from home plate. What is the initial speed of the ball, and how high does it rise? 4. Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the curve at point P. (a) r(t) = 4 cos ti + 4 sin tj, P(2 2,2 2) (6) r(t) = t^2i + tj + 4/3k, P(1, 1, 4/3) 5. Find the principal unit normal vector to the curve at the specified value of the parameter. r(t) = 3 cos ti + 3 sin tj, t = pi/4

Explanation / Answer

1)

a) given position r(t)=t2i +t j

velocity v(t)= r'(t)=2t i +1j

speed=|v(t)|=[(2t)2+12]

speed=|v(t)|=[4t2+1]

acceleration a(t)=v'(t)=2i +0j

b) given position r(t)=4ti +3cost j+3sint k

velocity v(t)= r'(t)=4i -3sint j+3cost k

speed=|v(t)|= [(4)2+(-3sint)2+(3cost)2]

speed=|v(t)|=[42+9(sint)2+9(cost)2]

speed=|v(t)|=[16+9]

speed=|v(t)|=25

speed=|v(t)|=5

acceleration a(t)=v'(t)=0i -3cost j-3sint k

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