Kinematics of Two-Dimensional

Motion

Kinematics of Two-Dimensional

Motion

• Positions, displacements, velocities, and accelerations are all vector quantities in two dimensions.

Position VectorsPosition Vectors

• Position is determined by using a Cartesian coordinate system.

• Convention uses a horizontal x-axis and a vertical y-axis.

Position VectorsPosition Vectors

• Position vector: r• tail at origin• head at object location• location of origin can be

arbitrarily assigned

Position VectorsPosition Vectors

• The coordinate system within which motion is measured or observed

• There is no absolute frame of reference.

Frame of ReferenceFrame of Reference

Change in position: Δrd = Δr = r2 – r1

r2 is the position at the endr1 is the initial position

DisplacementDisplacement

Displacement is the same regardless of the reference

frame used!

DisplacementDisplacement

Velocity and Speed in Two Dimensions

Velocity and Speed in Two Dimensions

Average velocity:

Average speed: v =

v =

sΔt

ΔrΔt

dΔt=

• shows the velocity of an object at any given moment

• points in the direction of movement at that instant

Instantaneous Velocity VectorInstantaneous Velocity Vector

• equal to the magnitude of the instantaneous velocity

Instantaneous Speed

Instantaneous Speed

v = |v|

• is often quite different from the magnitude of the average velocity

• Average speed equals average velocity only when s = |d|.

Average SpeedAverage Speed

Acceleration in Two Dimensions

Acceleration in Two Dimensions

• acceleration may involve:• change in magnitude • change in direction• change in both

Remember that acceleration is a change in velocity!

Acceleration in Two Dimensions

Acceleration in Two Dimensions

average acceleration vector is equal to the velocity

difference divided by the time interval:

a =v2 – v1

Δt ΔtΔv=

Acceleration in Two Dimensions

Acceleration in Two Dimensions

The direction of the average acceleration is always the

same direction as the velocity difference vector,

Δv.

Instantaneous Acceleration

Instantaneous Acceleration

• acceleration at a particular moment

• Its vector points in the same direction as the instantaneous velocity difference vector.

ProjectionsProjections

Projectiles Projectiles • any flying object that is

given an initial velocity, and is then influenced only by external forces, such as gravity

• includes objects that fall

Projectiles Projectiles • Trajectory: the path of a

projectile

Projectiles Projectiles • Ballistic trajectory: the

unpowered portion of a projectile’s path• gravitational force only• air resistance will be

disregarded

Horizontal Projections Horizontal Projections

• a motion in which an object is initially propelled horizontally and then allowed to fall in a ballistic trajectory

Horizontal Projections Horizontal Projections

• The kinematics of the horizontal and vertical components of motion are completely separate, but occur simultaneously.

Horizontal Projections Horizontal Projections

• The total velocity of a projectile at any time after launch is the vector sum of the horizontal and vertical velocity components.

Horizontal ComponentHorizontal Component

• The horizontal displacement is sometimes called the range.

• recall the first equation of motion:

v2x = v1x + axΔt

Horizontal ComponentHorizontal Component

• Since the horizontal acceleration is zero, we now have:

v2x = v1x

Horizontal ComponentHorizontal Component

• Similarly, the second equation of motion becomes:

x2 = x1 + vxΔt

dx = x2 - x1 = vxΔt

Horizontal ComponentHorizontal Component

• The third equation of motion becomes meaningless since it has a denominator of zero.

Vertical ComponentVertical Component

• downward acceleration is g = -9.81 m/s²

• For a horizontal projection, the initial vertical velocity (v1y) is zero.

Vertical ComponentVertical Component

• The final vertical velocity of a projectile is due solely to the amount of time it has to fall.

• positive direction is upward

Vertical ComponentVertical Component

• Equations of motion:

v2y = gyΔt

dy = ½gy(Δt)²

dy =v2y²

2gy .

Example 5-4 Example 5-4 • Find the time (Δt) using the

second equation (vertical)• Use the time to calculate

the range• Be careful with the units!

Frame of Reference Frame of Reference • motion may appear

different to different observers

Projection at an Angle Projection at an Angle • very common in the real

world• horizontal and vertical

accelerations the same as with a horizontal projection• ax = 0, ay = -g

Projection at an Angle Projection at an Angle • initial vertical velocity is no

longer zero• components of initial

vertical velocity:• v1x = v1 cos θv1 • v1y = v1 sin θv1

Projection at an Angle Projection at an Angle • These components can be

used in the original equations of motion—no need to memorize another set of equations!

Projectile MotionProjectile Motion

• It is possible to calculate the horizontal and vertical displacement components at any time during the projectile’s flight.

• These can also be graphed.

Projectile MotionProjectile Motion

• At the peak of its flight, the projectile’s vertical velocity is zero.

Projectile MotionProjectile Motion

• If air resistance, wind, etc. is ignored, several things can be noted:

Projectile MotionProjectile Motion

• The time it takes a projectile to go from a given height to its peak is the same time it takes to fall from its peak to that given height.

Projectile MotionProjectile Motion

• The trajectory is symmetrical.

• Vertical speed is the same at corresponding heights (but the direction has changed).

Projectile MotionProjectile Motion

• The equation of a ballistic trajectory is a quadratic function, and its graph (see Fig. 5-16) is a parabola.

Projectile MotionProjectile Motion

• Therefore, it is often good to know the quadratic formula:

-b ± b² - 4ac2a

x =

Projectile Motion Projectile Motion • In the real world, wind, air

resistance, and other factors will affect motion.

• To achieve maximum range ideally, a launch angle of 45° should be used.