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• 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
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.
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 • 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
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!
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 =