(1) A locomotive is traveling on the straight track as shown. At the instant when r = 400 m and θ = 75°, the tracking device records r = 20 m/s and θ= 0.02 rad/s2. Determine (a) the magnitudes of the velocity and acceleration of the locomotive, (b) θ and r at this
Ans: v = 40 m/s, a = 5.24 m/s2
= −0.0866 rad/s,
r = 0.381 m/s2
(2) The motion of a jet plane flying horizontally is being tracked by the radar located at O as shown. At the instant when r = 2500 m and θ = 60°, the tracking device records the values r = 100 m/s and θ= 0.002 rad/s2. Determine the value of r at this position.
Ans: 17.11 m/s2
(3) The ground radar is tracking the motion of an airplane that is flying in a straight line gaining altitude at a climb angle of β = 30°
shown. At a certain instant the recorded values indicate r = 1500 ft, θ = tan–1 (3/4),r = 360 ft/sec, and θ = –0.055 rad/sec2. Determine the value r for this position.
Ans: r = −4.57 ft/sec2
(4) The airplane dives down along a curved trajectory and at the bottom of the vertical loop it has a horizontal velocity v shown. At this lowest point, the radius of curvature of the loop is ρ = 1600 m and the speed of the plane is increasing at a rate of 5 m/s2. If the radar tracking indicates r = 500 m, θ = 60˚, and r = 100 m/s, determine (a) the speed v and (b) r and θ for this instant.
Ans: (a) v = 200 m/s (b) r = 84.2 m/s2, θ = 0.1549 rad/s2
(5) During a portion of the airplane’s vertical loop the angle of the velocity is β = 25° at point A and the radar tracking records values
of r = 1000 m, θ = 35˚, r =150 m/s, and r = 25 m/s2. If the radius of curvature at A is 600 m, determine θ and the tangential acceleration of the plane.
Ans: θ = −0.0736 rad/s2, at = −8.66 m/s2
(6) At an air show plane A flies along the indicated straight path while plane B executes a vertical loop shown. At the position under consideration plane A has a speed of 265 mi/hr that is increasing at a rate of 4 mi/hr/sec and the speed of plane B is 150 mi/hr that
remains constant. Determine the velocity and acceleration of plane B with respect to plane A for this instant.
Ans: / 355 15.43 ˆ ˆ B A v ij = − mi/hr, / 74.0 70.5 ˆ ˆ B A a ij = + ft/sec2
(7) For the instant represented car A is rounding the circular curve with a speed of 40 ft/sec and is speeding up at the rate of 2 ft/sec2, while car B on the straightaway is speeding up at the rate of 5 ft/sec2. Determine the relative acceleration of car A with respect to an observer in car B.
Ans: 1.130 0.5 ˆ ˆ − −i j ft/sec2