AP Physics Assignment – Advanced Kinematics

Reading            Physics for Scientists and Engineers   pp. 39 – 44, 53 – 64, 71 – 83, 87 – 91

Objectives/HW

 The student will be able to: HW: 1 Add or subtract vectors graphically and determine a vector's opposite. 1 2 Calculate the components of a vector given its magnitude and direction. 2 – 3 3 Calculate the magnitude and direction of a vector given its components. 4 – 5 4 Use vector components as a means of analyzing/solving 2-D motion problems. 6 – 7 5 Add or subtract vectors analytically (using trigonometric calculations). 8 – 10 6 Use vector addition or subtraction as a means of solving relative motion problems. 11 – 16 7 State the horizontal and vertical relations for projectile motion and use the same to solve projectile problems. 17 – 25 8 Use derivatives to determine speed, velocity, or acceleration and solve for extrema and/or zeros. 26 – 28 9 Use integrals to determine distance, displacement, change in speed, or change in velocity and solve for functions thereof given initial conditions. 29 – 32 10 Solve problems involving parametric equations that describe motion components 33 – 35

Homework Problems

1.      Given two vectors:  A = 10.0 cm, 30.0° and B = 20.0 cm, 135.0° determine the following:  (a) −A = ?, (b) −B = ?, (c) A + B = ?, (d) AB = ?, (e) BA = ?  Note: Use the graphical method to solve parts (c), (d), and (e) – this means to draw and measure the vectors using ruler and protractor.  Use the graphical method for this problem only!

2.      For each of the following vectors determine the x and y components:  (a) A = 5.00 m, 45.0°, (b) B = 8.0 m/s, 320°, (c) C = 9.80 m/s2, 270.0°

3.      A skateboarder coasts down a ramp at speed 6.0 m/s.  The ramp forms a 20.0° angle with the horizontal.  (a) Determine the vertical speed (or rate of descent).  (b) Determine the horizontal speed or (rate of forward progress).

4.      Given the two components, determine the magnitude and direction for the following:  (a) Ax = −15 km; Ay = 77 km, (b) Bx = −38.0 m/s; By = −28.0 m/s, (c) Cx = 950 km; Cy = 0

5.      A crewman runs at speed 8.0 m/s from port to starboard across the deck of a ship that is traveling south at 5.0 m/s.  Determine the resultant velocity of the crewman.

6.      Starting at elevation 3375 m, two climbers set out across a steep slope on a mountain side.  Francois climbs at velocity 1.5 m/s, 10.0° above horizontal, while Jacques takes a more direct route up the slope and climbs at velocity 0.80 m/s, 20.0° above horizontal.  (a) Which climber is ascending at a greater rate and by how much?  (b) If the rates are maintained, how much time does it take each climber to reach elevation 3495 m?

7.      In the cockpit of an airplane the vertical speed indicator shows an ascent rate of 9.0 m/s while the airspeed indicator reads 65 m/s.  (a) Relative to horizontal, in which direction is the plane moving?  (b) If these conditions are maintained, how far over level ground does the plane travel in 30.0 s?

8.      Add or subtract vectors as indicated and calculate the magnitude and direction of the result:
(a) (15 m, 0°) + (375 m, 180°)
(b) (25.0 m/s, 180.0°) − (18.0 m/s, 90.0°)
(c) (35 km, 90°) + (75 km, 0°) − (125 km, 270°)
(d) (120.0 km/h, 20.0°) + (40.0 km/h, 240.0°)

9.      Starting at a position of 100.0 km, 45.0° from Knoxburg, an airplane maintains a velocity of 350.0 km/h, 180.0° for 35.0 minutes.  Determine the final position.

10.  A car travels around a circle at a constant speed of 20.0 m/s.  A magnetic compass in the car indicates that its direction of travel changes from north to northeast in 4.00 seconds.  (a) Find the change in the car’s velocity during this interval.  (b) Determine the average acceleration.

11.  An airplane has airspeed 245 km/h and heading due north.  It encounters a wind of 64.0 km/h blowing southwest.  (a) Determine its groundspeed.  (b) Determine its course.

12.  A boater travels 25 m, east directly across a river that flows south at 2.0 m/s.  The boat’s speed through the water is 3.0 m/s.  (a) In what direction is the boat headed?  (b) Determine the time for the boater to cross the river.

13.  An airplane must travel 575 km west in 2.00 hours to its next destination.  If the aircraft encounters a wind of 50.0 km/h blowing southward, what must be the airspeed and heading?

14.  Two boats are traveling at the same speed through the water but traveling in opposite directions along a river.  A person on the riverbank measures one boat moving at 4.0 m/s and the other at 6.4 m/s.  (a) What is the speed of each boat relative to the water?  (b) How fast is the river moving relative to the shore?

15.  A cyclist traveling along a road at 12 km/h, 35° feels the wind on his face and it seems to be blowing directly from north to south.  However he can tell from nearby flags that the wind is actually blowing from west to east.  (a) What is the speed of the wind relative to earth?  (b) What is the speed of the wind relative to the cyclist?

16.  Two cars initially separated by 350 m travel in opposite directions on a two lane highway.  Car A has initial velocity 25 m/s, N and acceleration 0.50 m/s2, N; Car B has initial velocity 29 m/s, S and acceleration 0.20 m/s2, N.  (a) What is the acceleration of Car B from the perspective of car A?  (b) What is the initial velocity of Car B relative to Car A?  (c) How much time passes before the two cars pass one another?

17.  A boy throws a pebble horizontally with speed 12 m/s from a bridge that is 8.0 m above the water.  (a) How much time is the pebble in the air?  (b) How far is the pebble from the boy when it splashes?  (c) What is the speed of the pebble when it hits the water?

18.  A rifle is fired horizontally at a target 30.0 m away.  The bullet hits the target 1.50 cm below the point where it was aimed.  What is the “muzzle speed” of the bullet?

19.  A baseball is thrown horizontally toward home plate with a speed of 40.0 m/s.  The distance from the mound to home plate is 18.3 m.  (a) How much time does the batter have to react to and hit the ball?  (b) How much does the ball move vertically before reaching home plate?  (c) What is the velocity of the ball as it crosses home plate?

20.  A spud is launched from a potato cannon with velocity 55.0 m/s, 40.0° and flies out into the woods.  The sound of the spud hitting a tree is heard 2.00 seconds after launch.  (a) Where was the spud when it hit (i.e. what position)?  (b) What was the impact speed?

21.  A quarterback throws a football at 20.0 m/s, 50.0° which is caught in stride by a receiver running directly away from him at 9.0 m/s.  (a) Determine the time the ball is in the air.  (b) Determine the maximum height of the ball.  (c) Find the range of the ball.  (d) How far downfield was the receiver at the instant the ball was thrown?

22.  A football is kicked from a tee and travels downfield 60.0 yards before hitting the ground.  The football is airborne for 3.50 seconds (this is the “hang time”).  (a) What was the initial velocity of the ball, in mph, as it left the tee?  (b) What was the maximum height, in yards, above the field?  Try “American” units for this one:  g = 32.2 ft/s2, 1 yard = 3 feet, 1 mile = 5280 feet

23.  The class clown fires a spit wad from a straw with velocity 9.00 m/s, 60.0° from a position 2.00 m below the ceiling.  The projectile hits the ceiling instead of its intended target.  (a) Determine the impact velocity.  (b) If fired with the same initial speed what would be the maximum angle of launch that would avoid hitting the ceiling?

24.  A dart is thrown at velocity 10.0 m/s, 15.0° toward a target on a wall that is a 4.00 m away.  (a) Relative to its release point, how much higher or lower is the dart when it hits?  (b) What is the speed at impact?

25.  Suppose a volleyball player wants to serve the ball such that it just barely passes over the net.  Let x represent the horizontal distance to the net and y represent the height of the net relative to the launch point of the ball.  Derive an equation that gives the initial launch angle in terms of x and y.

26.  An object moves along the x-axis such that x = 2 t4 − 3 t2 + 5, where x = position in m and t = time in s.  (a) Determine the velocity function for this object.  (b) Determine the acceleration function for this object.  (c) What position reached by the object is closest to the origin?  (d) What is the maximum speed in a leftward direction?  (e) What is the greatest magnitude of acceleration that occurs in the first second of motion?

27.  A top fuel dragster can be modeled by d = 0.333 t4 − 7.5 t3 + 91 t2, where d = distance in ft and t = time in s.  The race is Ľ mile or 1320 ft.  At the point where it crosses the finish line determine (a) the time, (b) the speed, and (c) the rate of acceleration.  Note:  use the features of your calculator to solve for roots of polynomials – you do not have to solve explicitly.

28.  Consider a bungee jumper.  The motion that results from the pull of the bungee can be modeled by the following function:  v = −5 t3 + 22 t2 − 9.8 t − 20, where v = velocity in m/s and t = time in s.  This equation is valid only for a certain interval of time starting at the instant the bungee first starts to pull the falling person.  (a) Determine the acceleration as a function of time.  (b) Determine the time at which the bungee stops pulling (hint:  the person is once again in freefall at that point).  (c) Find the average acceleration that occurs while the bungee is pulling.  (d) Find the maximum upward acceleration caused by the bungee.

29.  (a) Assuming the initial position is y = 0, determine the function describing the position of the bungee jumper in the previous problem.  (b) Find the maximum amount that the bungee cord is stretched.

30.  An object moves along the y-axis in such a way that its acceleration in an upward direction is given by:  a = 5 t − 2, where a is acceleration in m/s and t is time in seconds.  At t = 4.0 s, the velocity of the object is 38 m/s upward and its position is 40.0 above the origin.  (a) Find the object’s velocity at t = 0.  (b) Find the object’s position at t = 0.  (c) Determine the minimum speed of the object and the position at which this occurs.

31.  Scientists speculate that bacteria may have survived being launched into space on bits of rock and soil released by the impact of an asteroid hitting the earth’s crust.  In such a scenario bacterium would be subject to a violent motion undergoing a change in acceleration from zero to 2.00 million g’s in 0.500 ms.  Assume the change in acceleration occurs at a constant rate and that the bacterium is initially at rest.  (a) Determine a(t).  (b) Determine v(t). (c) Determine x(t).  (d) Find the distance traveled and speed at t = 0.500 ms.

32.  Suppose the motion of a rocket can be modeled by the equation a = A + B t0.5.  Apply this model to a rocket launched from rest with initial acceleration 5.5 m/s2 upward that has an acceleration of 13 m/s2 upward 9.0 seconds after launch.  (a) Determine the values of A and B, including units.  (b) Find the distance traveled during the first 9.0 seconds.  (c) Find the speed at 9.0 seconds.

33.  An object moves in the xy-plane such that its position in meters is described by x(t) = 4t2 and y(t) = 3t4 − 10t2, where t is time in seconds.  (a) Determine the velocity of the object at t = 2 s.  (b) Determine the acceleration of the object at t = 2 s.  (c) At what point in time is the object moving in a purely horizontal direction.  (d) Find the equation of the path traveled by the object (i.e. find y as a function of x).

34.  An object initially at the origin has velocity with components described by vx(t) = 3 − 2t and vy(t) = 2 + 3t, where v is in m/s and t is in s.  (a) Determine the parametric equations that describe the position of the object as a function of time.  (b) Find the acceleration at t = 3 s.  (c) Determine the minimum speed of the object and the position at which this occurs.  (d) Describe the type of path followed and sketch the graph.

35.  A wombat is in the back of a flatbed truck unbeknownst to the driver.  At the instant the truck starts moving northward, the wombat runs eastward across the flat bed of the truck and off the edge.  The speed of the truck can be modeled by v = 8 t0.5.  Meanwhile the wombat’s motion is a constant acceleration of 3.0 m/s2 east (relative to the truck) as it crosses the 2.0 m width of the truck bed.  (a) Determine the velocity of the wombat as it flies off the edge of the truck bed.  (b) Determine the acceleration of the wombat at the same instant.  (c) Determine the equation describing the path followed by the wombat.  (d) Use the equation of path to determine how far forward the truck has moved by the time the wombat flies off.

1.      a. 10.0 cm, 210.0°
b. 20.0 cm, 315.0°
c. 19.9 cm, 106.0°
d. 24.6 cm, 338.2°
e. 24.6 cm, 158.2°

2.      a. Ax = 3.54 m, Ay = 3.54 m
b. Bx = 6.1 m/s, By = −5.1 m/s
c. Cx = 0, Cy = −9.80 m/s2

3.      a. 2.1 m/s
b. 5.6 m/s

4.      a. A = 78 km, 101°
b. B = 47.2 m/s, 216.4°
c. C = 950 km, 0°

5.      9.4 m/s, 32° S of W

6.      a. Jacques by 0.01 m/s
b. 7.3 min, 7.7 min

7.      a. 8.0° above horizontal
b. 1930 m

8.      a. 360 m, 180°
b. 30.8 m/s, 215.8°
c. 177 km, 64.9°
d. 93.0 km/h, 3.9°

9.      151 km, 152.1°

10.  a. 15.3 m/s, 337.5°
b. 3.83 m/s2, 337.5°

11.  a. 205 km/h
b. 102.8°

12.  a. 42°
b. 11 s

13.  292 km/h, 170.1°

14.  a. 5.2 m/s
b. 1.2 m/s

15.  a. 9.8 km/h
b. 6.9 km/h

16.  a. 0.30 m/s2, S
b. 54 m/s, S
c. 6.4 s

17.  a. 1.3 s
b. 17 m (8 m down, 15 m over)
c. 17 m/s

18.  542 m/s

19.  a. 0.457 s
b. drops 1.03 m
c. 40.3 m/s, 6.4° below horiz.

20.  a. 98.6 m, 31.2° from cannon
b. 45.0 m/s

21.  a. 3.13 s
b. 12.0 m above release
c. 40.2 m
d. 12 m

22.  a. 52.0 mph, 47.6°
b. 16.4 yds

23.  a. 6.47 m/s, 45.6°
b. 44.0°

24.  a. 0.232 m above release
b. 9.77 m/s

25.  θ = tan−1(2y/x)

26.  a. v = 8 t3 − 6 t
b. a = 24 t2 − 6
c. 3.88 m right of origin
d. 2.00 m/s
e. 18 m/s2

27.  a. 4.55 s
b. 488 ft/s (333 mph)
c. 60.0 ft/s2

28.  a. a = −15 t2 + 44 t − 9.8
b. t = 2.93 s
c. 11.7 m/s2 upward
d. 22.5 m/s2 upward

29.  a. y = −1.25t4 + 7.33t3 − 4.9t2 − 20t
b. 22.7 m

30.  a. 6.0 m/s, up
b. 21.3 m below origin
c. 5.6 m/s, 19.0 m below origin

31.  a. a = 3.92 × 1010 t
b. v = 1.96 × 1010 t 2
c. x = 6.53 × 109 t 3
d. 0.817 m, 4900 m/s

32.  a. A = 5.5 ms−2, B = 2.5 ms−2.5
b. 385 m
c. 94.5 m/s

33.  a. 58.2 m/s, 74.1°
b. 124 m/s2, 86.3°
c. 1.29 s
d. y = 0.188x2 − 2.5x

34.  a. x = 3tt2, y = 2t + 1.5t2
b. 3.61 m/s2, 123.7°
c. 3.61 m/s at origin
d. parabola w/ vertex on origin but tilted left by 33.7°

35.  a. 9.27 m/s, 68.1°
b. 4.78 m/s2, 51.1°
c. y = 3.935x0.75
d. 6.62 m