(Solved): The particle has the acceleration a(t) = -24k, initial velocity v(0) = i + j, and initial position ...

The particle has the acceleration a(t) = -24k, initial velocity v(0) = i + j, and initial position r(0) = 0. Find the velocity of the particle. (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) v(t): = Find the speed of the particle. (Express numbers in exact form. Use symbolic notation and fractions where needed.) v(t): = Find the position of the particle. (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) r(t) = Projectile Motion: Football In a field goal attempt on a flat field, a football is kicked at an angle of 30° to the horizontal with an initial speed of 61 ft/s. What horizontal distance does the football travel while it is in the air? (Use decimal notation. Give your answer to three decimal places. Assume 32 ft/s².) g= range ft To score a field goal, the ball must clear the cross bar of the goal post, which is 10 ft above the ground. What is the farthest from the goal post that the kick can originate and score a field goal? (Use decimal notation. Give your answer as a whole number.) X? ft A particle moves on the ellipse x² + 2 = 1 so that at time 1 ? 0, the position is given by the vector 2-1² 4t -i + 2+1² 2+12 r(1) (a) Find the velocity vector. (Use symbolic notation and fractions where needed. Give your answer as vector coordinates in the form (x(1), y(t)).) v(t) = (b) Is the particle ever at rest? No Yes (c) Give the coordinates of the point (x, y) that the particle approaches as increases without bound. (Use symbolic notation and fractions where needed. Give your answer as point coordinates in the form (*, *).) (X?o, Y?o) = Projectile Motion A projectile is fired at an angle of 30° to the horizontal with an initial speed of 560 m/s. What are its range, the time of flight, and the greatest height reached? (Use decimal notation. Give your answers as whole numbers. Assume g = 9.8 m/s².) range time of flight maximum height= m S m