Engineering Mechanics Dynamics. R.C. Hibbeler. 6th edition. Solution manual; manual & mathcad. December V0.1. Made by A.J.P.

Engineering Mechanics Dynamics R.C. Hibbeler 6th edition Solution manual; manual & mathcad December 2015. V0.1 Made by A.J.P. Schalkwijk

12.1 Rectilinear Kinematics: Continuous Motion

, example 12-1: The car in fig. 12-2 moves in a straight line such that for a short time it's velocity is defined by v=(9t^2 + 2t) ft/s, where t is in seconds. Determine its position and acceleration when t=3s. When t=0, s=0.

, example 12-2: A small particle is fired vertically downward into a fluid medium with an initial velocity of 60m/s. If the projectile experiences a deceleration which is equal to a=(-0.4v^3) m/s^2, where v is measured in m/s, determine the projectile's velocity and position 4s after it is fired.

, example 12-3: A boy tosses a ball in the vertical direction of the side of a cliff, as shown in fig. 12-4. If the initial velocity of the ball is 15m/s upward, and the ball is released 40m from the bottom of the cliff, determine the maximum height Sb reached by the ball and the speed of the ball just before it hits the ground. During the entire time the ball is in motion, it is subjected to a constant downward acceleration of 9.81m/s^2 due to gravity. Neglect the effect of air resistance.

, example 12-4: A metallic particle is subjected to the influence of a magnetic field such that it travels downward through a fluid that extends from plate A to plate B, fig 12-5. If the particle is released from rest at the midpoint C, s=100mm, and the acceleration is measured as a=(4s)m/s^2, where s is in meters, determine the velocity of the particle when it reaches plate B, s=200mm, and the time it needs to travel van C to B.

, example 12-5: A particle moves along a horizontal line such that it's velocity is given by v=(3t^2-6t)m/s, where t is the time in seconds. If it is initially located at the origin O, determine the distance traveled by the particle during the time interval t=0 to t=3.5, and the particle's average velocity and average speed during this time interval.

, problem 12-1: If a particle has an initial velocity of V0=12ft/sec to right, determine its position when t=10s, if a=2ft/sec^2 to the left. Originally s0=0.

, problem 12-2: 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/sec (55 mi/hr) when it hits the ground? Each floor 12ft higher than the one below it.

, problem 12-3: A particle is moving along a straight line such that its position is given by s=(4t-t^2) ft., where t is in seconds. Determine the distance travelled from t=0 to t=5s, the average velocity, and the average speed of the particle during this time interval.

, problem 12-5: A particle is moving along a straight line path such that it's position is defined by s=(10t^2+20)mm, where t is in seconds. Determine (a) the displacement of the particle during the time interval from t=1 s to t=5 s, (b) the average velocity of the particle during this time interval, and (c) the acceleration at t=1 s.

, problem 12-6: A ball is thrown vertically upward from the top of a ledge with an initial velocity of Va=35ft/sec. Determine (a) how high above the top of the cliff the ball will go before its stops at B, (b) the time Ta-b it takes to reach its maximum height, and (c) the total time Ta-c needed for it to reach the ground at C from the instant it is released.

, problem 12-7: A car, initially at rest, moves along a straight road with constant acceleration such that it attains a velocity of V=60ft/s when s=150ft. Then after being subjected to another constant acceleration, it attains a final velocity of V=100ft/s when s=325ft. Determine the average velocity and average acceleration of the car for the entire 325ft displacement.

, problem 12-9: When a train is travelling along a straight track at 2m/s, it begins to accelerate at a=(60v^-4)m/s^2, where V is in m/s. Determine the velocity V and the position of the train 3sec. after the acceleration.

, problem 12-10: A race car uniformly accelerates at 10ft/s^2 from rest, reaches a maximum speed of 60mi/h, and then decelerates uniformly to a stop. Determine the total elapsed time if the distance travelled was 1500ft.

, problem 12-11: A small metal particle passes through a fluid medium under the influence of magnetic attraction. The position of the particle is defined by s=(0.5t^3+4t)inch., where t is in seconds. Determine the position, velocity, and acceleration of the particle when t=3s.

, problem 12-13: A particle travels to the right along a straight path with a velocity v=[5/(4+s)]m/s, where s is in meters. Determine its position when t=6s if s=5m when t=0.

, problem 12-14: The velocity of a particle traveling along a straight line is v=(6t-3t^2)m/s, where t is in seconds. If s=0 when t=0, determine the particle's deceleration and position when t=3s. How far has the particle traveled during the 3-s time interval, and what is the average speed?

, problem 12-17: At the same instant, two cars A and B start from rest at a stop line. Car A has a constant acceleration of aa=8m/s^2, while car B has an acceleration of ab=(2t(3/2))m/s^2, where t is in seconds. Determine the distance between the cars when A reaches a velocity of Va=120km/h.

, problem 12-18: A particle moves along a straight path with an acceleration of a=(5/s)m/s^2, where s is in meters. Determine the particle's velocity when s=2m if it is released from rest when s=1m.

, problem 12-19: A particle moves with accelerated motion such that a=-k*s, where s is the distance from the starting point and k is a proportionality constant which is to be determined. When s=2ft the velocity is 4ft/s, and when s=3.5ft the velocity is 8ft/s. What is s when v=0?

, problem 12-21: A particle moving along a straight line is subjected to a deceleration a=(-2v^3)m/s^2, where v is in m/s. If it has a velocity v=8m/s and a position s=10m when t=0, determine its velocity and position when t=4s.

, problem 12-22: The acceleration of a rocket travelling upward is given by a=(6+0.02s) m/s^2, where s in meters. Determine the rocket's velocity when s=2km and the time needed to reach this elevation. Initially, v=0 and s=0 when t=0.

, problem 12-23: Two trains are traveling in opposite directions on parallel tracks. Train A is 150m long and has a speed which is twice as fast as train B, which is 250m long. Determine the speed of each train if a passenger in train A observes that train B passes in 4s.

, problem 12-25: The juggler maintains the motion of three balls, such that each rises to a height of 4ft. If two balls are in the air at any one time, determine the time the third ball must remain in her hand after the first ball is thrown.

, problem 12-26: The juggler throws a ball into the air 4ft above her hand. How much time will elapse before she must catch it at the same elevation from which she threw it? What would be the elapsed time if she threw it 8 ft into the air?

, problem 12-27: When two cars A and B are next to one another, they are traveling in the same direction with speeds Va and Vb, respectively. If B maintains its constant speed, while A begins to decelerate at aa, determine the distance between the cars at the instant A stops.

, problem 12-29: When a particle falls through the air, it's initial acceleration a=g diminishes until it is zero, and thereafter it falls at a constant velocity Vf. If this variation of the acceleration can expressed as a=(g/vf^2)(vf^2-v^2), determine the time needed for the velocity to become V<Vf. Initially the particle falls from rest.

12.2 Rectilinear Kinematics: Erratic Motion

, example 12-6: A car moves along a straight line path such that its position is described by the graph shown in fig. 12-9a. Construct the v-t and a-t graphs for the time period 0<t<30s.

, example 12-7: A rocket sled starts from rest and travels along a straight track such that it accelerates at a constant rate for 10s and then decelerates at a constant rate. Draw the v-t and s-t graphs and determine the time t' needed to stop the sled. How far has the sled traveled?

, problem 12-8: The v-s graph describing the motion of a motorcycle is shown in fig. 12-15a. Construct the a-s graph of the motion and determine the time needed for the motorcycle to reach the position s=400ft.

, problem 12-31: If the position of a particle is defined as s=(5t-3t^2) ft, where t is in seconds, construct the s-t, v-t and a-t graph for 0 < t < 10 s.

, problem 12-33: The speed of a train during the first minute of its motion has been recorded as follows: Plot the v-t graph, approximating the curve as straight line segments between the given points. Determine the total distance traveled.

, problem 12-34: The s-t graph for a train has been determined experimentally. From the data, construct the v-t and a-t graphs for the motion.

, problem 12-35: Two cars start from rest side by side at the same time and position and race along a straight track. Car A accelerates at 4 ft/ s^2 for 35 s and then maintains a constant speed. Car B accelerates at 10 ft/s^2 until reaching a speed of 45mi/h and then maintains a constant speed. Determine the time at which the cars will again be side by side. How far has each car traveled? Construct the v-t graphs for each car.

, problem 12-37: From experimental data, the motion of a jet plane while traveling along a runway is defined by the v-t graph. Construct the s-t and a-t graphs for the motion.

, problem12-38: The car travels along a straight road according to the v-t graph. Determine the total distance the car travels until it stops when t=48sec. Also plot the s-t and a-t graphs. Page 23, problem 12-39: The snowmobile moves along a straight course according to the v-t graph. Construct the s-t and a-t graphs for the same 50 s time interval. When t=0, s=0.

, problem 12-41: The v-t graph for the motion of a car as it moves along a straight road is shown. Construct the s-t graph and determine the average speed and the distance traveled for the 30 s time interval. The car starts from rest at s=0.

, problem 12-42: A particle starts from rest and is subjected to the acceleration shown. Construct the v-t graph for the motion, and determine the distance traveled during the time interval 2s < t < 6s.

, problem 12-43: An airplane lands on the straight runway, originally traveling at 110ft/s when s=0. If it is subjected to the decalerations shown, determine the time t' needed to stop the plane and construct the s-t graph for the motion:

, problem 12-45: The a-t graph for a car is shown. Construct the v- t graph if the car starts from rest at t=0. At what time t' does the car stop? Page 24, problem 12-46: A race car starting from rest travels along a straight road and for 10s has the acceleration shown. Construct the v-t graph that describes the motion and find the distance traveled in 10s.

, problem 12-47: The boat is originally traveling at a speed of 8 m/s when it is subjected to the acceleration shown in the graph. Determine the boat's maximum speed and the time t when it stops.

, problem 12-49: The a-s graph for a race car moving along a straight track has been experimentally determined. If the car starts from rest, determine its speed when s=50 ft, 150ft and 200ft, respectively.

, problem 12-51: The jet plane starts from rest at s=0 and is subjected to the acceleration shown. Construct the v-t graph and determine the time needed to travel 500ft.

, problem 12-53: The v-s graph for the car is given for the first 500ft of its motion. Construct the a-s graph for 0<s<500ft. How long does it take to travel the 500ft distance? The car starts at s=0 when t=0.

, problem 12-54: The a-s graph for a boat moving along a straight path is given. If the boat starts at s=0 when v=0, determine its speed when it is at s=75ft and 125ft respectively. Use Simpson's rule with n=100 to evaluate v at s=125ft.

12.3 General Curvilinear motion

, example 12-9: At any instant the position of the kite in fig. 12-18a is defined by the coordinates x=30t and y=9t^2 ft, where t is given in seconds. Determine (a) the equation which describes the path and the distance of the kite from the boy when t=2s, (b) the magnitude and direction of the velocity when t=2, and (c) the magnitude and direction of the acceleration when t=2sec.

, example 12-10: The motion of a bead B sliding down along the spiral path shown in fig. 12-19 is defined by the position vector r={0.5sin(2t)i + 0.5cos(2t)j - 0.2tk}m, where t is given in seconds and the arguments for sine and cosine are given in radians (pi rad = 180deg). Determine the location of the bead when t=0.75s and the magnitude of the bead's velocity and acceleration at this instant.

, example 12-11: A ball is ejected from the tube, shown in fig 12-21 with a horizontal velocity of 12 m/s. If the height of the tube is 6m, determine the time needed for the ball to strike the floor and the range R.

, example 12-12: A ball is thrown from a position 5 ft above the ground to the roof of a 40ft high building, as shown in fig 12-22. If the initial velocity of the ball is 70ft/s, inclined at an angle of 60deg from the horizontal, determine the range or horizontal distance R from the point where the ball is thrown to where it strikes the roof.

, example 12-13: When a ball is kicked from A as shown in fig. 12-23, it just clears the top of the wall at B as it reaches its maximum height. Knowing that the distance from A to the wall is 20m and the wall is 4m heigh, determine the initial speed at which the ball was kicked. Neglect the size of the ball.

, problem 12-55: If x=1-t and y=t^2, where x and y are in meters and t is in seconds, determine the x and y components of velocity and acceleration and construct the path y=f(x).

, problem 12-57: If the position of a particle is defined by its coordinates x=4t^2 and y=3t^3, where x and y are in meters and t is in seconds, determine the x and y components of velocity and acceleration and construct the path y=f(x).

, problem 12-58: The position of a particle is defined by If, where t is in seconds, determine the particles velocity v and acceleration a at the instant t=1 sec. Express v and a as Cartesian vectors.

, problem 12-59: If the velocity of a particle is v(t)={0.8t^2i + 12t^0.5j + 5k}m/s, determine the magnitude and coordinate direction angles a,b and c of the particle's acceleration when t=2s. Page 39, problem 12-61: A particle travels along the curve from A to B in 2s. It takes 4s for it to go from B to C and then 3s to go from C to D. Determine its average velocity when it goes from A to D.

, problem 12-62: A particle moves with curvilinear motion in the positive x-y plane such that the y component of motion is described by y=7t^3, where y is in feet and t is in seconds. When t=1s, the particle's speed is 60ft/s. If the acceleration of the particle in the x direction is zero, determine the velocity of the particle when t=2s.

, problem 12-63: A car traveling along the road has the velocities indicated in the figure when it arrives at it points A, B and C. If it takes 10s to go from A to B, and then 15s to go from B to C, determine the average acceleration between points A and B and between points A and C.

, problem 12-66: The flight path of the helicopter as it takes off from A is defined by the parametric equations x=(2t^2)m and y=(0.04t^3)m, where t is the time in seconds after takeoff. Determine the distance the helicopter is from point A and the magnitude of its velocity and acceleration when t=10s.

, problem 12-67: A particle is moving along the curve y=x-(x^2/400). If the velocity component in the x direction is Vx=2ft/s, determine the magnitude of the particle's velocity and acceleration when x=20ft.

, problem 12-69: A particle moves along a hyperbolic path X^2/16 - y^2=28. If the x component of its velocity is always Vx=4m/s, determine the magnitude of its velocity and acceleration when it is at point (32m,6m).

, problem 12-74: A basketball is tossed from A at angle of 30deg from the horizontal. Determine the speed Va at which the ball is released in order to make the basket B. With what speed does the ball pass through the hoop?. Neglect the size of the ball in the calculation.

, problem 12-78: The centre of the wheel is traveling at 60ft/s. If it encounters the transitions of two rails, such that there is a drop of 0.25inch. at the joint between the rails, determine the distance s to point A where the wheel strikes the next rail.

, problem 12-82: A boy at A throw's a ball 45deg from the horizontal such that it strikes the slope at B. Determine the speed at which the ball is thrown and the time of flight.

, problem 12-83: A frog jumps upward, perpendicular to the incline, with a velocity of Va=10ft/sec. Determine the distance R where it strikes the plate at B.

12.6: Curvilinear motion: normal and tangential components

, problem 12-93: A particle is moving along a curved path at a constant speed of 60ft/s. The radii of curvature of the path at points P and P' are 20 and 50 ft, respectively. If it takes the particle 20 sec to go from P to P', determine the acceleration of the particle at P and P'. Page 51, 12-94: A car travels along a horizontal curved road that has a radius of 600m. If the speed is uniformly increased at a rate of 2000km/ h^2, determine the magnitude of the acceleration at the instant the speed of the car is 60km/h.

, problem 12-95: A boat is traveling along a circular path having a radius of 20m. Determine the magnitude of the boat's acceleration if at a given instant the boat's speed is v=5 m/s and the rate of increase in the speed is dv/dt =2 m/s^2. Page 51, problem 12-97: A car moves along a circular track of radius 100ft such that it's speed for a short period of time 0 <= t <= 4s is v=3(t+t^2) ft/s, where t is in seconds. Determine the magnitude of its acceleration when t= 2s. How far has the car traveled in 2s?

, problem 12-99: A race car has an initial speed of V0=15m/s when s=0. If it increases its speed along the circular track at the rate of at=(0.4s) m/s^2, where s is in meters, determine the normal and tangential components of the car's acceleration when s=100m.