Two Dimensional Kinematics

1. Two Dimensional 
Kinematics

2. Position and Velocity Vectors y z A y x z x 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 + +

3. 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)

4. Components of Instantaneous Velocity The instantaneous velocity can have three different 
components: x, y, and z. Each component is shown below, Vector representation:

5. 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.

6. Components of Instantaneous Acceleration Vector representation:

7. Given    . Find  and .

8. Given . Find and  .

9. 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. vy vx vx vx vy v vy vx vx vy

10. Velocity of a Projectile vy vx vx vx vy v vy vx vx 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.

11. Velocity of a Projectile vy vx vx vx vy v vy vx vx 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).

12. 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.

13. 1 Which of the following statements are 
true regarding projectile motion? A is constant 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.

14. 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. Which of these indicates the direction of the speed, 
if any, of the marble at point Y? A B v C D E None

15. 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. Which of the following indicates the direction 
of the net force on the marble at Point X? A B v C D E

16. Time to fall from apex When a projectile is thrown in the horizontal direction v H

17. 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

18. To Find Maximum Height Because at the highest point the 
vertical component of velocity is 
zero. (time to attain maximum height)

19. To Find Maximum Displacement vy vx vx vx vy v vy vx vx vy

20. To find angle between the velocities vx vy vy vx vy vy

21. 5 At what angle will a projectile have the 
greatest vertical displacement? A 0 B 30 C 45 D 60 E 90

22. 6 At what angle will a projectile have the greatest 
horizontal displacement? A 0 B 30 C 45 D 60 E 90

23. 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.

24. 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.

25. 8 A car is driving with decreasing velocity on a 
curved path. Which diagram shows the correct 
direction for the velocity and acceleration? v v A B a a C D v v a a E v a

26. 9 A car is driving with constant velocity on a curved 
path. Which diagram shows the correct direction 
for the velocity and acceleration? v B A v a a C D v v a a E v a

27. 10 A car is driving with increasing velocity on a 
curved path. Which diagram shows the correct 
direction for the velocity and acceleration? v A v B a a C D v v a a E v a

28. Uniform Circular Motion (centripetal acceleration) If we plug the equation for the 
velocity into the acceleration 
equation we get:

29. Uniform Circular Motion Centripetal Acceleration P2 P1 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

30. Uniform Circular Motion Centripetal 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)

31. 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

32. 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

33. 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

34. 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. arad arad atan arad atan arad arad arad atan atan

35. 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.

36. 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

37. 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 1200 km/h B 1300 km/h 500 km/h C 700 km/h D 1700 km/h E 2500 km/h

38. 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