Kinematics in Two Dimensions

Chapter Three

Scalar Vs. Vector

Scalar– Magnitude only

Vector– Magnitude and direction– Vector Addition (trigonometry)

Vector Notation

v = speed

v (or v )= velocity

Example One

A student walks 10 km due east, and then 4 km due north. How far is she from her starting point?

Tip-to-Tail Method

Correct method

tail

tip

Resultant

Example Two

A student walks 50 m due west, turns and walks 50 m northeast. How far are they away from the starting point?

Example Three

A sasquatch walks 100 m at an angle 20o south of west. He then walks 40 m due north, then 65 m at an angle of 35o north of west.

a)Sketch his path

b)Calculate the distance and angle from the origin.

Vector Resolution

A car travels 50 m/s at an angle 30o north of east. Draw and measure its x and y component vectors.

30o

V

East

North

Vector Resolution: Trigonometry

sin = opposite = o

hypotenuse h

cos = adjacent = a

hypotenuse h

tan = opposite = o

adjacent a

h

a

o

Vector Resolution: Ex. 1

A car travels 500 km at an angle 30o north of east. Calculate its x and y displacement.

30o

500 km

East

North

Vector Resolution : Ex. 2

A car travels 300 km at an angle of 22.0o W of N. Calculate the x and y-components of the vector.

(Dx = -112 km, Dy = 278 km)

Vector Resolution : Ex. 3

A car travels 40.0 m/s at an angle of 56.0o S of E. Calculate the x and y-components of the velocity.

(Dx = 22.4 m/s, Dy = -33.2 m/s)

Vector Resolution : Ex. 4

A mailman travels 300 m at an angle of 25o N of E, then 100 m at an angle 50o N of E. Calculate the total (resultant) displacement and angle. (393 m, 31.1o north of east)

25o

B=100 m

East

North

A=300 m50o

25o

B=100 m

East

North

A=300 m50o

Ax

Ay

Bx

By

Vector Resolution : Ex. 5

A mail carrier drives 22.0 km north. She then drives 47.0 km in a direction 60.0o S of E. What is her displacement from the post office?

(Ans: 30.0 km, -38.5o)

Vector Resolution : Ex. 6

A plane travels due east for 620 km, 65o S of E for 440 km, and then 53o S of W for 550 km. What is the displacement from the airport?

(Ans: 960 km, -61o)

Vector Resolution : Ex. 7

A bird flies 5.00 km due south, then 3.00 km at an angle of 25o north of east, then 4.00 km at an angle of 45o south of east. Calculate the magnitude and direction of the net displacement.

(8.59 km, 49.8o S of E)

Vector Resolution : Ex. 8

A bicycle travels at • 3.0 m/s, 45o N of E• then 5.0 m/s, 45o W of N• then 2.0 m/s, 60o N of E.

a)Calculate the resultant (net) velocity and the angle.

b)Can you express your answer in i and j notation?

(7.40m/s, 93.2o)

A person travels 5.00 km southwest, then 7.00 km 35.0o north of east, then 3.00 km due east. Calculate the net displacement and angle. Can you express your answer in i and j notation?

Which will hit the ground first, a bullet fired from a gun, or just dropped from the same height?

Projectile Motion

• Definition – motion in both horizontal and vertical direction

• Examples: Arrows, baseballs, bullets

• Trajectory – curved path taken by objects in projectile motion

• Thrown objects fall just like dropped objects

Vectors are Independent

1. x and y components are independent of one another.

2. x component NEVER changes

3. y component changes due to gravity

Vx never changes

Vy increases from the acceleration of gravity

Can Superman throw a baseball all the way around the earth, turn around, and catch it at the same height?

Example 1

Assume the wagon is moving at a constant velocity. Can the girl in the wagon through the ball straight up and catch it, or will she pass under it?

Example 2

A bug is sitting on your dashboard when you start and then get the car up to a constant speed. If the bug takes off, will he slam into the back window of your car?

If the kid in the tree lets go just as the water-balloon gun is fired, will he get hit?

Example 3

A movie stuntman drives a motorcycle off a 50.0 m cliff.

a)Calculate the time of flight. (3.19 s)

b)Calculate how fast must it travel to hit the ground 90.0 m from the base of the cliff? (vxo=28.2 m/s)

c)Calculate the vertical speed at which the bike hits the ground. (31.2 m/s)

Projectile Motion

Mr. Fredericks throws a discus from a 20.0 m hill with a horizontal velocity of 8.6 m/s.a)Calculate the vertical velocity of the discus as it hits the ground. (19.8 m/s)b)What is the horizontal velocity as it hits the ground?c)Calculate the time of flight. (2.02 s)d)Calculate how far the discus lands from Mr. Fredericks. (17.4 m)

Projectile Motion

A cliff diver runs from the top of a 15.0 m cliff and hits the water 30.0 m from the base of the cliff.

a)Calculate the time the diver was in the air. (1.75 s)

b)Calculate the vertical velocity at which they hit the water. (17.1 m/s)

c)Calculate the initial horizontal velocity. (17.1 m/s)

Example 4a

A football is kicked at an angle of 37.0o with a velocity of 20.0 m/s.

a) Calculate the maximum height of the football. (7.35 m)

b) Calculate the total time that the ball is in the air. (2.45 s)

c) Calculate how far the ball travels horizontally (39.2 m)

d) What is the ball’s velocity at the highest point?e) What is the acceleration of the ball at the

highest point?

 A pirate is attacking a gold bearing ship. The pirate's cannon can shoot projectiles at a speed of 70.0m/s, the cannons are tilted at 30° up.

a. Calculate the x and y components of velocity. (60.6 m/s, 35 m/s)

b.Calculate the total time in the air. (7.14 s)

c. Calculate how close the pirate ship must get for the cannon ball to hit the other ship. (432 m)

A girl kicks a soccer ball at an unknown angle and velocity. The ball travels 50.0 m downfield in 2.80 s.

a)Calculate vx (17.9 m/s)

b)Calculate the time to reach the peak height (1.40s)

c)Calculate the peak height (9.60 m)

d)Calculate vy (13.7 m/s)

e)Calculate the initial velocity and angle to the ground. (22.5 m/s, 37.4o)

A student kicks a dodge ball at an unknown angle and velocity. The ball travels 25.0 m the gym in 1.07 s.

a)Calculate vx (23.4 m/s)

b)Calculate the time to reach the peak height (0.535 s)

c)Calculate vy (5.24 m/s)

d)Calculate the peak height (1.40 m)

e)Calculate the initial velocity and angle to the ground. (24.0 m/s, 12.6o)

Example 5

Suppose Mario can hit a golf ball at 60.0 m/s. What angle should he strike the ball to hit a green 320 m away?

Range Formula

R =(v2sin2 g

=30.3o and 59.7o)

Range

What angle should any projectile be fired to achieve maximum range?

R = (v2sin2 g

We want sin2 to be the maximum value (1)

2 = 90o

Note how 60o and 30o produce the same range

A football is kicked 50.0 m at a speed of 24.0 m/s. Calculate the launch angle.

(29.1o or 60.9o)

Mr. Fredericks throws a lazy physics student with a velocity of 500.0 m/s and an angle of 45.0o. Where does the student land, in pain?

Example 7

A boy kicks a football from a height of 1 m. The football was kicked at an angle of 37o and a velocity of 20.0 m/s.

a) Calculate the vx and vy (16.0, 12.0 m/s)

b) Calculate the time to peak height (1.22 s)

c) Calculate the peak height (remember that it was kicked from 1 m) (8.35 m)

d) Calculate the time from peak height to the ground. (1.30 s)

e) Calculate the total time in the air (2.52 s)

f) Calculate the x-distance (40.3 m)

A baseball is thrown at 10.0 m/s from the top of a 5.00 m hill at an angle of 45.0o.

a. Calculate the peak height. (7.55 m)

b.Calculate the time until it reaches peak height. (0.72 s)

c. Calculate the total time in the air.(1.96 s)

d.Calculate the total horizontal distance the ball can travel. (13.9 m)

A ball is thrown from ground level to a roof that is 7.00 m above the ground. You stand 5.00 m from the wall of the building, and throw the ball at 14.2 m/s and an angle of 80.8o.

a)Calculate the x and y components of the initial velocity (vx = 2.27 m/s,voy = 14.0 m/s)

b)Calculate the time it takes to reach the peak height. (1.42 s)

c)Calculate the peak height. (10.0 m)

d)Calculate the total time (2.20 s)

Damien is thrown from ground level to a roof that is 15.00 m above the ground. You stand 10.00 m from the wall of the building, and throw Damien at 50.0 m/s and an angle of 75.0o.

a)Components of velocity (12.9 m/s and 48.3 m/s)

b)Time up (4.93 s)

c)Peak height (119 m)

d)Time down (4.61 s)

Relative Velocity

A car is traveling on a highway at 65 mph. What speed does it seem to be traveling if you are:

a) Stopped at the entrance ramp to the highway?

b) Traveling in the same direction at 55 mph?

c) Traveling in the opposite direction at 60 mph?

SPEED LIMIT

65

Suppose a boat wishes to go straight across the Susquehanna river (which flows south). Which direction should they head?

Flow of river

Boat starts here

Rel. Velocity: Example 1

A boat crosses straight across a river at a speed of 1.85 m/s. The current of the river is 1.20 m/s west.

a) At what angle did the boat start out?

b)What was the straight velocity?

1.20 m/s

1.85 m/s

Rel. Velocity: Example 2

Suppose that same boat headed straight across the stream.

a) What is the boat’s velocity (2.21 m/s)

b) How far downstream would it be after the trip? (the river is 110 m wide). (72 m)

c) How far did the boat actually travel? (132 m)

1.20 m/s

1.85 m/svr

Rel. Velocity: Example 3

You are travelling in a plane at 250 km/h due east. A wind blows at 50.0 km/h from the southwest.

a)Calculate the resultant velocity (and angle) of the plane. (288 km/h, 7.1o N of E)

b)Calculate the distance the plane travelled in 40 minutes (192 km)

A plane travels at 200 km/h due north. However, a 100 km/h wind coming from the northeast blows.

a)Calculate the resulting velocity of the plane with

respect to the ground. (147 km/h)b)Calculate the direction of the planes motion.

(28.7o W of N)c)Calculate how far the plane travelled north in

half an hour. (65 km)d)Calculate how far the plane travelled total in half

an hour. (73.5 km)

A pilot wishes to fly a distance of 300. km and on a bearing of 30.0o North of West. There is a wind blowing from the South-East at 100. km/h. The plane has an airspeed of 300. km/h. Calculate:

a)the ground speed of the plane (398 km/hr)

b)the pilot’s heading (angle) (33.7o N of W)

c)Calculate how long it will take the plane to travel 300 km. (45 minutes)

2. 20 blocks, 37o (S of E)

4.24.95, 41.10o below the x-axis (S of E)

6.b) -14.0, 19.9 c) 24.3, 54.8o above the x-axis

8. 5.84, 33.1o N of W

16. a) 1.90 m/s2 down b) 18.8 s

20.y = 44 m, x = 4.8 m 22. vox = 7.0 m/s

24. vo = 13 m/s

26. 2.46 s

28. 22 m

30. a) vo = 9.39 m/s b) 0.80 m

32. a) 1.13 m b) o = 0.54o 62. 46.3, 31.6o

32. a) 1.13 m b) o = 0.54o

36.a) 14.6 s b) 1.22 km c) 83.9 m/s

d) -79.9 m/s d) 116 m/s e) 43.6o below horizontal

40. 10.5 m/s in direction of ship’s motion, 6.5 m/s in direction of ship’s motion

42. vsc = 29 m/s vsg = 14 m/s down

44.0.00600 h = 21.6 s

46. 8.13o West of South

50. 0.90 m/s

52. = 53o 62. 46.3, 31.6o