Dynamics Kinematics

Kinematics of a Particle WWW.CLEAR-MATH.COM A bicyclist starts from rest and after traveling along a straight path a distance of 20 m reaches a speed of 30 km/h. Determine his acceleration if it is constant. Also, how long does it take to reach the speed of 30 km/h?

A car starts from rest and reaches a speed of 80 ft/s after traveling 500 ft along a straight road. Determine its constant acceleration and the time of travel.

A particle travels along a straight line such that in 2 s it move from an initial position s A = +0.5 m to a position sB = -1.5 m. Thin in another 4 s it moves from sB to sC = +2.5 m. Determine the particle’s average velocity and average speed during the 6 s time interval

Course Engineering Mathematics 1 ก่อนสอบ 3 สัปดาห์ เรียน 20 ชม. สอนเนื้อหาได้ 70 % ค่าเรียน 1,500 บาท/คน เทอม 2 เปิดสอน Engineering Mathematics 2 – 3 , Engineering Mechanics , Calculus 2 , Linear Algebra Summer เปิดสอน Engineering Mechanics , Engineering Mathematics 1 – 3 , Calculus 1 - 2

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

1

Kinematics of a Particle WWW.CLEAR-MATH.COM From approximately what floor of a building must a car be dropped from an at rest position so that it reaches a speed of 80.7 ft/s ( 55 mi/h ) when it hits the ground? Each floor is 12 ft higher than the one below it ( Note: you may want to remember this when traveling 55 mi/h )

The acceleration of a particle along a straight line is defined by a = ( 2t – 9 ) m/s2 where t is in seconds. At t = 0, s = 1 m and v = 10 m/s. When t = 9 s, determine (a) the particle’s position (b) the total distance traveled (c) the velocity

Course Engineering Mathematics 1 ก่อนสอบ 3 สัปดาห์ เรียน 20 ชม. สอนเนื้อหาได้ 70 % ค่าเรียน 1,500 บาท/คน เทอม 2 เปิดสอน Engineering Mathematics 2 – 3 , Engineering Mechanics , Calculus 2 , Linear Algebra Summer เปิดสอน Engineering Mechanics , Engineering Mathematics 1 – 3 , Calculus 1 - 2

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

2

Kinematics of a Particle WWW.CLEAR-MATH.COM Two particles A and B state from rest at the origin s = 0 and move along a straight line such that a A = ( 6t – 3 ) and aB = ( 12t2 – 8 ) ft/s2 where t is in seconds. Determine the distance between them when t = 4 s and the total distance each has traveled in t = 4 s

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

3

Kinematics of a Particle WWW.CLEAR-MATH.COM An airplane starts from rest, travels 5000 ft down a runway and after uniform acceleration takes off with a speed of 162 mi/h. It then climbs in a straight line with a uniform acceleration of 3 ft/s2 until it reaches a constant speed of 220 mi/h. Draw the s – t , v – t and a – t graphs that describe the motion

The v-s graph for an airplane traveling on a straight runway is shown. Determine the acceleration of the plane at s = 100 m and s = 150 m. Draw the a-s graph

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

4

Kinematics of a Particle WWW.CLEAR-MATH.COM The girl always throws the toys at an angle of 30 from point A as shown. Determine the time between throws so that both toys stride the edges of the pool B and C at the same instant. With what speed must she throw each toy?

The balloon A is ascending at the rate v  12 km/h and is being carried horizontally by the wind at v  20 km/h. If a ballast bag is dropped from the balloon at the instant h = 50 m, determine the time needed for it to strike the ground. Assume that the bag was released from the balloon with the same velocity as the balloon. Also, with what speed does the bag strike the ground? A

W

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

5

Kinematics of a Particle WWW.CLEAR-MATH.COM The man stands 60 ft from the wall and throws a ball at it with a speed 50 ft/s. Determine the angle at which he should release the ball so that it strikes the wall at the highest point possible. What is this height? The room has a ceiling height of 20 ft.

Course Engineering Mathematics 1 ก่อนสอบ 3 สัปดาห์ เรียน 20 ชม. สอนเนื้อหาได้ 70 % ค่าเรียน 1,500 บาท/คน เทอม 2 เปิดสอน Engineering Mathematics 2 – 3 , Engineering Mechanics , Calculus 2 , Linear Algebra Summer เปิดสอน Engineering Mechanics , Engineering Mathematics 1 – 3 , Calculus 1 - 2

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

6

Kinematics of a Particle WWW.CLEAR-MATH.COM The ball is throw from the tower with a velocity of 20 ft/s as shown. Determine the x and y coordinates to where the ball strikes the slope. Also, determine the speed at which the ball hits the ground

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

7

Kinematics of a Particle WWW.CLEAR-MATH.COM The ball is kicked with an initial speed v  8 m/s at an angle   40 with the horizontal. Find the equation of the path y  f ( x) and then determine the ball’s velocity and the normal and tangential components of its acceleration when t = 0.2 s 

A

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

A

8

Kinematics of a Particle WWW.CLEAR-MATH.COM If a particle’s position is described by the polar coordinates r  4(1  sin t ) m and   (2e ) rad where t is in seconds and the argument for the sine is in radians, determine the radial and tangential components of the particle’s velocity and acceleration when t = 2 s t

A particle moves in the x-y plane such that its position is defined by r  {2ti  4t j } ft where t is in seconds. Determine the radial and tangential components of the particle’s velocity and acceleration when t = 2 s 2

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

9

Kinematics of a Particle WWW.CLEAR-MATH.COM The slotted link is pinned at O and as a result of the constant angular velocity of 3 rad/s it drives the peg P for a short distance along the spiral guide r  (0.4 ) m where  is in radians. Determine the radial and transverse components of the velocity and acceleration of P at the instant    3 rad

Determine the displacement of the block at B it A is pulled down 4 ft

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

10

Kinematics of a Particle WWW.CLEAR-MATH.COM The pulley arrangement shown is designed for hoisting materials. If BC remains fixed while the plunger P is pushed downward with a speed of 4 ft/s, determine the speed of the load at A

If block A is moving downward at 6 ft/s while block C is moving down at 18 ft/s, determine the relative velocity of block B with respect to C

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

11

Kinematics of a Particle WWW.CLEAR-MATH.COM If block A of the pulley system is moving downward with a speed of 4 ft/s while block C is moving up at 2 ft/s, determine the speed of block B

At the instant shown. Cars A and B are traveling at speeds of 30 mi/h and 20 mi/h, respectively. If B is increasing its speed by 1200 mi/h2 while A maintains a constant speed, determine the velocity and acceleration of B with respect to A

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

12

Kinematics of a Particle WWW.CLEAR-MATH.COM At the instant shown, the bicyclist at A is traveling at 7 m/s around the curve on the race track while increasing his speed at 0.5 m/s2 . The bicyclist at B is traveling at 8.5 m/s along the straight-a-way and increasing his speed at 0.7 m/s2 . Determine the relative velocity and relative acceleration off A with respect to B at this instant

Course Engineering Mathematics 1 ก่อนสอบ 3 สัปดาห์ เรียน 20 ชม. สอนเนื้อหาได้ 70 % ค่าเรียน 1,500 บาท/คน เทอม 2 เปิดสอน Engineering Mathematics 2 – 3 , Engineering Mechanics , Calculus 2 , Linear Algebra Summer เปิดสอน Engineering Mechanics , Engineering Mathematics 1 – 3 , Calculus 1 - 2

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

13

Kinematics of a Particle WWW.CLEAR-MATH.COM At a given instant the football player at A throws a football C with a velocity of 20 m/s in the direction shown. Determine the constant speed at which the player at B must run so that he can catch the football at the same elevation at which it was thrown. Also calculate the relative velocity and relation acceleration of the football with respect to B at the instant the catch is made. Player B is 15 m away from A when A starts to throw the throw the football

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

14

Kinematics of a Particle WWW.CLEAR-MATH.COM Point A oscillates with an acceleration a  144(20  x) where a and x are expressed in in/s2 and inches, respectively. Knowing that the system starts at time t = 0 with v = 0 and x = 19 in , determine the position and the velocity of A when t = 0.2 s

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

15

Kinematics of a Particle WWW.CLEAR-MATH.COM Steep safety ramps are built beside mountain highways to enable vehicles with defective brakes to stop safety. A truck enters a 250 m ramp at a high speed v and travels 180 m in 6 s at constant deceleration before its speed is reduced to v 2 . Assuming the same constant deceleration, determine (a) the additional time required for the truck to stop (b) the additional distance traveled by the truck 0

0

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

16

Kinematics of a Particle WWW.CLEAR-MATH.COM Car A is parked along the northbound lane of a highway and car B is traveling in the southbound lane at a constant speed of 96 km/h. At t = 0, A starts and accelerates at a constant rate a while at t = 5 s , B begins to slow down with a constant deceleration of magnitude a 6 . Knowing that when the cars pass each other x = 90 m and v  v , determine (a) the acceleration A (b) when the vehicles pass each other (c) the distance between the vehicles at t = 0 A

A

A

B

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

17

Kinematics of a Particle WWW.CLEAR-MATH.COM Block C starts from rest and moves downward with a constant acceleration. Knowing that after 12 s the velocity of block A is 456 mm/s, determine (a) the acceleration of A, B and C (b) the velocity and the change in position of block B after 8 s

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

18

Kinematics of a Particle WWW.CLEAR-MATH.COM Slider block A starts with an initial velocity at t = 0 and a constant acceleration of 9 in/s 2 to the right. Slider block C starts from rest at t = 0 and moves to the right with constant acceleration. Knowing that at t = 2 s , the velocity of A and B are 14 in/s to the right and 1 in/s to the left, respectively, determine (a) the acceleration of B and C (b) the initial velocity of A and B (c) the initial velocity of portion E of the cable

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

19

Kinematics of a Particle WWW.CLEAR-MATH.COM A parachutist is in free fall at a rate of 180 ft/s when he opens his parachute at an altitude of 1900 ft. Following a rapid and constant deceleration he then descends at a constant rate of 44 ft/s from 1800 ft to 100 ft , where he maneuvers the parachute into the wind to further slow his descent. Knowing that the parachutist lands with a negligible downward velocity, determine (a) the time required for the parachutist to land after opening his parachute (b) the initial deceleration

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

20

Kinematics of a Particle WWW.CLEAR-MATH.COM Car A is traveling at 70 km/h when it enters a 50 km/h speed zone. The driver of car A decelerates at a rate of 5 m/s 2 until reaching a speed of 50 km/h which she then maintains. When car B which was initially 19 m behind car A and traveling with a constant speed of 72 km/h enters the speed zone ,its driver decelerates at a rate of 6 m/s 2 until reaching a speed of 48 km/h. Knowing that the driver of car B maintains a speed of 48 km/h, determine (a) the closest that car B comes to car A (b) the time at which car A is 22 m in front of car B

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

21

Kinematics of a Particle WWW.CLEAR-MATH.COM

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

22

Kinematics of a Particle WWW.CLEAR-MATH.COM A baseball pitching machine ‘throws’ baseballs with a horizontal velocity. Knowing that height varies between 788 mm and 1068 mm, determine (a) the range of values of velocity (b) the values of  corresponding to h = 788 mm and h = 1068 mm

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

23

Kinematics of a Particle WWW.CLEAR-MATH.COM A golfer hits a golf ball with an initial velocity of 48 m/s at an angle of 25 with the horizontal. Knowing that the fairway slopes downward at an average angle of 5 , determine the distance d between the golfer and point B where the ball first lands 



In slow pitch softball the underhand pitch must reach a maximum height of between 6 ft and 12 ft above the ground. A pitch is made with an initial velocity of magnitude 43 ft/s at an angle of 33 with the horizontal. Determine (a) if the pitch meets the maximum height requirement (b) the height of the ball as it reaches the batter 

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

24

Kinematics of a Particle WWW.CLEAR-MATH.COM Small wheels attached to the ends of rod AB roll along two surfaces. Knowing that at the instant shown the velocity (b) the v of wheel B with respect to wheel A is perpendicular to rod AB, determine (a) the relative velocity v velocity v of wheel B A

B A

B

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

25

Kinematics of a Particle WWW.CLEAR-MATH.COM Slider block B starts from rest and moves to the right with a constant acceleration of 300 mm/s 2. Determine (a) the relative acceleration of portion C of the cable with respect to slider block A (b) the velocity of portion C of the cable after 2 s

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

26

Kinematics of a Particle WWW.CLEAR-MATH.COM Pin P moves at a constant speed of 8 in/s in a counterclockwise sense along a circular slot which has been milled in the block A as shown. Knowing that the block moves up the incline at a constant speed of 4.8 in/s, determine the magnitude and the direction relative to the xy axes of the velocity of pin P when (a)   30 (b)   135 

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)



27

Kinematics of a Particle WWW.CLEAR-MATH.COM The rotation of rod OA about O is defined by the relation   0.5e sin 3 t where  and t are expressed in radians and seconds, respectively. Collar B slides along the rod so that its distance from O is r  0.2  1.92t  6.72t  6.4t where r and t are expressed in meters and seconds, respectively. When t = 0.5 s, determine (a) the velocity of the collar (b) the acceleration of the collar (c) the acceleration of the collar relative to the rod 0.8t

2

3

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

28

Kinematics of a Particle WWW.CLEAR-MATH.COM Pin C is attached to rod BC and slides freely in the slot of rod OA which rotates at the constant rate  . At the instant when   60 , determine (a) r and  (b) r and  . Express your answers in terms of d and  

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

29

Kinematics of a Particle WWW.CLEAR-MATH.COM Block B starts from rest and moves downward with a constant acceleration. Knowing that after slider block A has moved 400 mm its velocity is 4 m/s, determine (a) the accelerations of A and B (b) the velocity and the change in position of B after 2 s

Course Engineering Mathematics 1 ก่อนสอบ 3 สัปดาห์ เรียน 20 ชม. สอนเนื้อหาได้ 70 % ค่าเรียน 1,500 บาท/คน เทอม 2 เปิดสอน Engineering Mathematics 2 – 3 , Engineering Mechanics , Calculus 2 , Linear Algebra Summer เปิดสอน Engineering Mechanics , Engineering Mathematics 1 – 3 , Calculus 1 - 2

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

30

Kinematics of a Particle WWW.CLEAR-MATH.COM An oscillating garden sprinkler which discharges water with an initial velocity of 8 m/s is used to water a vegetable garden. Determine the distance to the farthest point B that will be watered and the corresponding angle when (a) the vegetables are just beginning to grow (b) the height h of the corn is 1.8 m

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

31

Kinematics of a Particle WWW.CLEAR-MATH.COM Knowing that at the instant shown block A has a velocity of 8 in/s and an acceleration of 6 in/s2 both directed down the incline, determine (a) the velocity of block B (b) the acceleration of block B

086 – 323 – 3270 (พี่ชัย) , 085 – 961 – 7666 (พี่พงษ์)

32