General Physics (calculus based) Class Notes

Dr. Rakesh Kapoor, M.Sc., Ph.D.

Former Faculty-University of Alabama at Birmingham, Birmingham, AL 35294

Motion in Two and Three Dimensions


In this chapter we will study motion in a plane (two dimensions) and motion in space (three dimensions):

The following vectors will be defined for two- and three- dimensional motion:

Position and Displacement

Average and Instantaneous velocity

Average and instantaneous acceleration

We will consider in detail projectile motion and uniform circular motion as examples of two dimensional motion.

Finally we will consider relative motion, i.e. the transformation of velocities between two reference systems which move with respect to each other with constant velocity.


In three dimensional space, a particle is located by a position vector MotionInTwoAnd3D_1.gif.

Vector MotionInTwoAnd3D_2.gif extends from a reference point (origin of a coordinate system) to the particle.


In three dimension MotionInTwoAnd3D_4.gif has three components and in unit vector notation MotionInTwoAnd3D_5.gif is written as


x, y and z are three components of the position vector MotionInTwoAnd3D_7.gif.


Watch the motion of three disks from initial position MotionInTwoAnd3D_8.gif (click play).

All the three disks moved (displaced) through different paths to a final position MotionInTwoAnd3D_10.gif.

How displacement is computed?

Displacement is computed by subtracting initial position MotionInTwoAnd3D_11.gif from final position MotionInTwoAnd3D_12.gif.


Although all the three disks (red, blue and green) starting from same initial position MotionInTwoAnd3D_14.gif follow different paths to reach the  final position MotionInTwoAnd3D_15.gif, but all three paths connecting the two points correspond to the same displacement vector MotionInTwoAnd3D_16.gif

Displacement vector is drawn as an arrow (purple arrow) connecting the initial point with final point with its arrowhead at final position

In unit vector notation, displacement vector  MotionInTwoAnd3D_17.gif is computed as


We can also rewrite displacement vector  MotionInTwoAnd3D_19.gif by substituting MotionInTwoAnd3D_20.gif, MotionInTwoAnd3D_21.gifand MotionInTwoAnd3D_22.gif


Average Velocity

When a particle moves from a position MotionInTwoAnd3D_24.gif to MotionInTwoAnd3D_25.gif in time interval Δt, the average velocity MotionInTwoAnd3D_26.gif is given as


Average velocity MotionInTwoAnd3D_28.gif in terms of component is written as



Direction of average velocity vector MotionInTwoAnd3D_31.gif (green arrow) is always along the displacement vector MotionInTwoAnd3D_32.gif as shown in above figure.

Instantaneous Velocity

Instantaneous velocity MotionInTwoAnd3D_33.gif is the velocity at some instant.

When we shrink the value of time interval Δ t such that Δt0, average velocity MotionInTwoAnd3D_34.gif approaches to instantaneous velocity MotionInTwoAnd3D_35.gif.

(reduce the time interval Δt and watch)


In unit vector form




What will be the direction of instantaneous velocity?

Let us try to reduce the value of Δ t in the following 2D simulation by sliding the Δt bar towards left hand side.

Red line passing through initial position (1) of the particle is the tangent to the trajectory (path) of the particle.

What we observed?

When the value of Δ t→0, the MotionInTwoAnd3D_41.gif becomes MotionInTwoAnd3D_42.gif and its direction is along the tangent to the trajectory at the particle position at that instant.

Press the play button in the following simulation and observe how the direction of velocity vector MotionInTwoAnd3D_43.gif changes as the particle moves along the trajectory.

The direction of instantaneous velocity MotionInTwoAnd3D_45.gif of a moving particle at any point on the path of the particle is always along the tangent (red line) to the path (trajectory) at that point or position.

Checkpoint - 1

Following figure shows a circular path taken by a particle. Instantaneous velocity of the particle is given in the figure.
For given velocity, through which quadrant is the particle moving at that instant if it is traveling (a) clockwise and (b) counterclockwise around the circle? For both cases, draw  MotionInTwoAnd3D_46.gif on the figure.

Draw the velocity vector. (Click, hold and drag the velocity vector to any location)
To bring it back to center, click at the center.

Hint: The direction of instantaneous velocity MotionInTwoAnd3D_48.gif of a moving particle at any point on the path of the particle is always along the tangent to the path at that point or position.
Move the velocity vector and find out at which point velocity vector is along the tangent to the circle?

Average Acceleration and Instantaneous Acceleration

When a velocity of a particle changes from MotionInTwoAnd3D_49.gif to MotionInTwoAnd3D_50.gif in time interval Δt, the average acceleration MotionInTwoAnd3D_51.gif is given as


Average acceleration MotionInTwoAnd3D_53.gif in terms of component is written as


When we shrink the value of Δ t to zero, the MotionInTwoAnd3D_55.gif becomes instantaneous acceleration (or acceleration) MotionInTwoAnd3D_56.gif at that instant.


In unit vector form




Relation between direction of velocity and acceleration

Watch the motion of red disk (click play).

We can observe that the velocity vector (green) and acceleration vector (orange) change with time.

Red line is tangent to the trajectory and blue arrow is position vector.

What we observed?

MotionInTwoAnd3D_61.gif is always along the tangent line (red) to the trajectory at that instant.

At the turning point observe, MotionInTwoAnd3D_62.gif and MotionInTwoAnd3D_63.gif are at right angle to each other.

Projectile Motion

What is a projectile?

Any motion satisfying following two conditions is called projectile motion.

When an object moves ONLY under the influence of acceleration due to gravity g.


Its initial velocity MotionInTwoAnd3D_65.gif must have a finite horizontal component MotionInTwoAnd3D_66.gif.




Click the play button and observe the trajectory of projectile motion.

How velocity components change as the particle move?

Horizontal component remained constant.

Vertical component changed.

In projectile motion, the horizontal motion and the vertical motion are independent of each other; means neither motion affects the other.

Projectile Motion Analyzed

The Horizontal Displacement

In case of a projectile motion, horizontal component of acceleration MotionInTwoAnd3D_70.gif, therefore horizontal displacement


The Vertical Displacement

In case of a projectile motion, vertical component of acceleration MotionInTwoAnd3D_72.gif, therefore vertical displacement


The Horizontal Range

The horizontal range MotionInTwoAnd3D_74.gif of the projectile is the horizontal distance the projectile has traveled when it returns to its initial (launch) height or when MotionInTwoAnd3D_75.gif.

How to compute range?

Range is the horizontal displacement in the time "t" when object comes back to its original Vertical position or vertical displacement becomes zero.


Once we know time "t", we get range R as below.


Eliminating t, from above two equations yields


Using trigonometric identity sin 2θ=2sinθ cosθ, we get


Horizontal range R will be maximum when sin 2θ=1, or MotionInTwoAnd3D_80.gif

In the following simulation play and change the angle.

Find out at what angle is the range maximum?

The horizontal range R is maximum for a launch angle of MotionInTwoAnd3D_82.gif.

Checkpoint 2

In the above projectile simulation observe the direction of velocity components as it moves.
How MotionInTwoAnd3D_83.gif and MotionInTwoAnd3D_84.gif behave?
Suppose at a certain instant, a fly ball has velocity  MotionInTwoAnd3D_85.gif (the x axis is horizontal, the y axis is upward, and MotionInTwoAnd3D_86.gif is in meters per second).
Has the ball passed its highest point?

Uniform Circular Motion

A particle moving with constant speed v in a circular path is said to be in uniform circular motion. (See the following simulation.)

Is velocity MotionInTwoAnd3D_88.gif constant or changing?

Is magnitude of velocity (speed) MotionInTwoAnd3D_89.gif constant or changing?

Time Period

The period of revolution T is given as


where r is the radius of the circle and v is the speed (magnitude of velocity) of the rotating object.

The position vector of a particle moving around a circle or a circular arc at constant (uniform) speed is given as



θ is a function of time and its value is 2π in one period of revolution T, so θ(t) can be written as


By substituting value of T we get


By substituting value of θ(t), the position vector can be written as


Velocity and acceleration can be computed as



Magnitude of  MotionInTwoAnd3D_98.gif is found to be


We know that




What is the direction of MotionInTwoAnd3D_102.gif ?

In circular motion, acceleration MotionInTwoAnd3D_104.gif always points  radially inward.

This radially inward acceleration associated with uniform circular motion is called a centripetal (meaning “center seeking”) acceleration.

Checkpoint 3

An object moves at constant speed along a circular path in a horizontal xy plane, with the center at the origin. When the object is at x=-2m, its velocity is MotionInTwoAnd3D_105.gif. Give the object’s (a) velocity and (b) acceleration at  y=-2m.

Relative Motion in One Dimension

When two cars are moving side by side with same velocity, for the drivers of each car, the other car is stationary while for an observer on the road, both cars are moving.

In other words, the velocity of a particle depends on the reference frame of observer measuring the velocity.

Consider two frames A and B with a red disk moving in Frame B.

When Frame A and Frame B both are stationary, the observer in frame A and another observer in  frame B measure the position of red disk P (Click Play to move disk P).

Here position measured by both the observers will be same.


Suppose Frame B is moving with constant velocity MotionInTwoAnd3D_109.gif in stationary frame A.  (Reset, select Frame B moving and click Play).

Now the relation between the positions measured by two observers at any instant will be given as


Where MotionInTwoAnd3D_111.gif and MotionInTwoAnd3D_112.gif are the positions of particle measure by observer in frame A and in frame B respectively, MotionInTwoAnd3D_113.gif is the position of B measured by observer in frame A.

Relation between the velocities can be given as



If frame B is moving with a constant velocity, the relation between acceleration is given as.


Because MotionInTwoAnd3D_117.gif is constant, therefore its derivative is MotionInTwoAnd3D_118.gif.


Observers on different frames of reference that move at constant velocity relative to each other will measure the same acceleration for a moving particle.

Relative Motion in Two Dimensions

In two dimensions suppose a particle P is moving in frame B and frame B is moving with velocity MotionInTwoAnd3D_120.gif with respect to frame A (Click Reset and then play).

The relation between the position measurements of particle P by observer in frame A (MotionInTwoAnd3D_122.gif) and by an observer in frame B  (MotionInTwoAnd3D_123.gif) will be given as


Where MotionInTwoAnd3D_125.gif and MotionInTwoAnd3D_126.gif are the positions of particle measure by observer in frame A and frame B respectively, MotionInTwoAnd3D_127.gif is the position of origin of frame B from origin of frame A.


Relation between the velocities can be given as



Similarly the relation between acceleration is given as.



If frame B is moving with a constant velocity MotionInTwoAnd3D_133.gif, its derivative will be zero therefore MotionInTwoAnd3D_134.gif.