: t³ - 61² + The motion of a particle is defined by the relation x = 9t + 5, where x is expressed in feet and t in seconds. Determine (a) when the velocity is zero, (b) the position, acceleration, and total distance traveled when t = 5 s.

At a certain point in the reentry of the space shuttle into the earth's atmosphere, the total acceleration of the shuttle may be represented by two components. One component is the gravitational acceleration g = 9.67 m/s² at this altitude. The second component equals 10.66 m/s² due to atmospheric resistance and is directed opposite to the velocity. The shuttle is at an altitude of 49.6 km and has reduced its orbital velocity of 28300 km/h to 16160 km/h in the direction = 1.23°. For this instant, calculate the radius of curvature of the path and the rate i at which the speed is changing. Answers: p= i = i i km m/s²

A motorcycle travels along a straight line with a velocity [ v = ( 2t² – 8 ) m/s ], where ( t ) is in seconds. When ( t = 1 s ), the motorcycle is located ( 10 m ) to the right of the origin. Determine: a- The displacement from ( t = 0 ) to ( t = 10 s ). b- The distance traveled by the motorcycle for the same time interval. c- The average velocity at this time interval. d- The average speed at this time interval. e- The acceleration at ( t = 5 s ).

A helicopter with a mass of 10,000 kg takes off straight up at t=0. The thrust of the engine is defined given as a function of time: FT= 95 0.18t- (kN). Where t is in units of seconds. How fast is it rising and what is its elevation at t=4s?