Two Dimensional Kinematics

Position and Velocity Vectors

If an object starts out at the origin and moves to point A, its displacement can be represented by a position vector.

x

y

z

x

yz A

x y z+ +

Position and Velocity Vectors

As an object moves from one point in space to another, the average velocity of its motion can be described as the displacement of the object over the time it takes to move.

(average velocity vector)

To find the instantaneous velocity (the velocity at a specific point in time) it requires the time interval to be so small that it can effectively be reduced to 0 which can be represented as a limit expression.

(instantaneous velocity vector)

Components of Instantaneous Velocity

The instantaneous velocity can have three different components: x, y, and z.

Each component is shown below,

Vector representation:

Acceleration Vector

Acceleration is the rate at which the velocity is changing, and the average acceleration can be found by taking the difference of the final and initial velocity and dividing it by the time it takes for that event to occur.

Just as we can find the velocity at a specific point in time, we can also find the instantaneous acceleration using a limit expression.

Components of Instantaneous Acceleration

Vector representation:

Given    . Find and .

Given . Find and  .

Projectile Motion

Have you ever thrown an object in the air and watched the trajectory it follows?

The path the object travels is a parabolic path.

vx

vy

vx

vx

vx

vxv

vy

vy

vy

Velocity of a Projectile

vx

vy

vx

vx

vx

vxv

vy

vy

vy

To explain projectile motion in the vertical direction we can use our knowledge of throwing a ball straight up into the air. We know that eventually the acceleration due to gravity will eventually stop the ball and make it move back towards the Earth. At its apex the ball stops moving in the vertical direction, so for a projectile this would be the same.

Velocity of a Projectile

vx

vy

vx

vx

vx

vxv

vy

vy

vy

To account for the projectile's motion in the horizontal direction we imagine a case where a block moves to the right with a velocity, v. In the absence of a resistant force (e.g. air resistance or friction) we can state that the block moves with a constant velocity (note that gravity does not affect the projectile's horizontal motion).

Position of an object in projectile motion

The position of the projectile with respect to its starting position can be represented with minor changes to the kinematics equation:

Horizontal position: Vertical Position:

From these equations we can determine the maximum horizontal distance, the maximum height reached by the projectile, the time to reach its highest point, and the time it hits the floor again.

1 Which of the following statements are true regarding projectile motion?

is constantA

B Acceleration is +g when the object is rising and -g when falling.

C In the absence of friction the trajectory will depend on the object's mass as well as its initial and launch angle.

D The velocity of the object is zero at the point of maximum elevation.

E The horizontal motion is independent of the vertical motion.

2 A marble is shot and follows a parabolic path shown below. Air resistance is negligible. Point Y is the highest point on the path.

A

B

C

D

v

Which of these indicates the direction of the speed, if any, of the marble at point Y?

E None

3 A marble is show and follows a parabolic path shown above. Air resistance is negligible. Point Y is the highest point on the path.

A

B

C

D

E

v

Which of the following indicates the direction of the net force on the marble at Point X?

H

v

Time to fall from apex

When a projectile is thrown in the horizontal direction

4 Two cannon balls are launched simultaneously off a cliff. The two cannon balls have different masses and different initial velocities. Which will strike the ground first?

A The heaviest one

B The lightest one

C The slowest one

D The fastest one

E They will both strike the ground at the same time

To Find Maximum Height

Because at the highest point the vertical component of velocity is zero.

(time to attain maximum height)

To Find Maximum Displacement

vx

vy

vx

vxvx

vxv

vy

vy

vy

To find angle between the velocities

vy

vx

vy

vx

vyvy

5 At what angle will a projectile have the greatest vertical displacement?

A 0B 30

C 45

D 60

E 90

6 At what angle will a projectile have the greatest horizontal displacement?

A 0

B 30

C 45

D 60

E 90

7 Which angles will have the same horizontal displacement?

A 0 and 90

B 30 and 60

C 0 and 45

D 35 and 60 E 30 and 90 F None of the two angles above will have the same

displacement.

Moving in a Circular Path

constant speed decreasing speed increasing speed

When an object moves in a circle with constant speed and its acceleration is perpendicular to the velocity this is called Uniform Circular Motion.

8 A car is driving with decreasing velocity on a curved path. Which diagram shows the correct direction for the velocity and acceleration?

A B

C Dv

a

v

a

v

a

v

a

v

a

E

9 A car is driving with constant velocity on a curved path. Which diagram shows the correct direction for the velocity and acceleration?

A B

v

a

v

a

v

a

v

a

v

a

C D

E

10 A car is driving with increasing velocity on a curved path. Which diagram shows the correct direction for the velocity and acceleration?

A B

C Dv

a

v

a

v

a

v

a

v

a

E

Uniform Circular Motion

(centripetal acceleration)

If we plug the equation for the velocity into the acceleration equation we get:

Uniform Circular MotionCentripetal Acceleration

P1

P2

Knowing that the triangles are similar, we can use ratios of corresponding sides, therefore:

To find the instantaneous velocity, we first have to come up with a representation for the average acceleration as before

Uniform Circular MotionCentripetal Acceleration

To find the instantaneous acceleration we have to take a limit expression of the average acceleration.

The limit expression will give us the velocity at a certain point in time, this velocity is the same as v1

(centripetal acceleration)

11 If a ball is swung in a circle of a radius of 1 m with a velocity of 5 m/s what would be the centripetal acceleration?

A 5 m/s2

B 0.2 m/s2

C 25 m/s2

D 0.04 m/s2

E 10 m/s2

12 If a ball is swung in a circle of radius 9 m and its centripetal acceleration was 1 m/s2. What would be its velocity?

A 3 m/s

B 9 m/s

C 81 m/s

D 18 m/s

E √3 m/s

13 If an object is moving in a circle with a velocity of 15 m/s and has a centripetal acceleration of 45 m/s2. What would be its radius?

A 5 m

B 1/3 m

C 3 m

D 10 m

E 15 m

arad

atan

arad

arad

aradarad

arad

atanatan

atan

Non-Uniform Circular Motion

When you are on a roller coaster and you come to a circular loop, your velocity is not constant. As you approach the top of the loop your velocity decreases and as you come back down your velocity increases. You still have a radial acceleration but now there is a tangential acceleration which is perpendicular to your radial acceleration.

Relative Velocity

When you are riding in a car and you look out a window what do you see?

If there is another car moving along side you with the same velocity relative to you, the other car appears to stand still, but with respect to the ground both of you are moving.

If another car is moving with velocity 2v with respect to the ground, then with respect to your car its moving with a velocity of v.

Relative Velocity

If a plane is flying through the air and enters a crosswind it will have a velocity straight and one perpendicular to it.

Vplane

Vcrosswind

14 A plane is moving with a constant speed of 1200 km/h and during part of its flight there is a cross wind blowing at 500 km/h. What is the net velocity during this portion of its flight?

A 1600 km/h

B 1300 km/h

C 700 km/h

D 1700 km/h

E 2500 km/h

1200 km/h

500 km/h

15 Two kids are on a boat capable of a maximum speed of 10 kilometers per hour in water, and wish to cross a river 2 kilometers wide to a point directly across from their starting point. If the speed of the water in the river is 9 kilometers per hour how much time is required for the crossing?

A 0.05 hrs

B 0.45 hrs

C 1 hr

D 10 hrs

E Not possible