Name: __________________________ Date: _____________ 1. The position vector r t 7t , 4 cos t , 4sin t describes the path of an object moving in space. Find the speed s t of the object. 2. Use the given acceleration function and initial conditions to find the position at time t = 1. a (t ) 6i \u000e 10 j \u000e 10 k , v (0) 0, r (0) 2 j 3. A baseball, hit 3 feet above the ground, leaves the bat at an angle of 60q and is caught by an outfielder 3 feet above the ground and 200 feet from home plate. What is the initial speed of the ball? Round your answer to two decimal places. 4. The quarterback of a football team releases a pass at a height of 7 feet above the playing field, and the football is caught by a receiver 46 yards directly downfield at a height of 3 feet. The pass is released at an angle of 75q with the horizontal. Find the maximum height of the football. Round your answer to one decimal place. 5. Find the unit tangent vector T(t) and then use it to find a set of parametric equations for the line tangent to the space curve given below at the given point. r (t ) -5t i + 4t 2 j \u000e 3tk , t 4 6. Verify that the space curves intersect at the given values of the parameter. Find the angle between the tangent vectors at the point of intersection. r (t ) t 3 \u0010 3t , 2t 2 , t , t u( s ) -3s , s ,sin s , s 0 0 7. Find the principle unit normal vector to the curve given below at the specified point. r (t ) t i \u000e 6 j, t t 2 Page 1 8. Find aT at time t 1 for the plane curve r t 7t 2 i \u000e 4tj . Round your answer to three decimal places. 9. Find aN at time t 1 for the plane curve r t 3t 2 i \u000e 4tj . Round your answer to three decimal places. S for the space curve r t 9 cos ti \u000e 9sin tj \u000e 5tk . Round your 3 answer to three decimal places. 10. Find aT at time t 11. Use the result, "the tangential and normal components of acceleration for a projectile fired at an angle T with the horizontal at an initial speed of v0 are aT \u0010 32 v0 sin T \u0010 32t v0 2 cos 2 T \u000e v0 sin T \u0010 32t 2 and aN 32v0 cos T v0 2 cos 2 T \u000e v0 sin T \u0010 32t 2 respectively.", to find the normal component of acceleration for a projectile fired at an angle of 45q with the horizontal at an initial speed of 120 feet per second. What is the component when the projectile is at its maximum height? 12. Find the length of the plane curve given below. r (t ) 5t 2 i \u000e 4t 2 j, [0,9] 13. Find the length of the space curve given below. r (t ) 5 t i \u000e 7 cos t j \u000e 7 sin t k , [0,6] 14. Find the curvature K of the curve given below. r (t ) t i \u000e 6t 2 j \u000e 4t k 15. Find the curvature K, where s is the arc length parameter. 9 5s 5s r ( s ) 5cos sk i \u000e 5sin j \u000e 706 706 706 Page 2 16. Find the curvature K of the curve r t ti \u000e t 2 j \u000e t3 k at the point P 2, 4, 2 . Round 4 your answer to three decimal places. 17. Find the curvature of the plane curve y decimal places. 5 x 2 \u000e 3 at x 18. Find the radius of curvature of the plane curve y answer to three decimal places. -1 . Round your answer to three 5 x 2 \u000e 6 at x -2 . Round your 19. The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the 4 3 curvature. A car moving on the path y x (x and y are measured in miles) can safely 3 32 3 9 go 25 miles per hour at 2, . How fast can it go at , ? Round your answer to 3 2 2 two decimal places. 20. A 5700-pound vehicle is driven at a speed of 35 miles per hour on a circular interchange of radius 95 feet. To keep the vehicle from skidding off course, what frictional force must the road surface exert on the tires? Round your answer to one decimal place. 21. Find the unit tangent vector T, the principal unit normal N, and the binormal vector B at t t1 . 1 2 1 r (t ) t i \u000e tj \u000e t 3k ; t1 2 a) 2 3 22. Find the unit tangent vector T, the principal unit normal N, and the binormal vector B at t t1 . S a) r (t ) e 7 t cos(2t )i \u000e e7 t sin(2t ) j \u000e e7 t k ; t1 3 Page 3