Kinematics of Two-Dimensional Motion

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

Position is determined by using a Cartesian coordinate system. Convention uses a horizontal x-axis and a vertical y-axis. Position Vectors

Position vector: r tail at origin head at object location location of origin can be arbitrarily assigned Position Vectors

The coordinate system within which motion is measured or observed There is no absolute frame of reference. Frame of Reference

Change in position: Δ r d = Δ r = r 2 – r 1 r 2 is the position at the end r 1 is the initial position Displacement

Displacement is the same regardless of the reference frame used! Displacement

Velocity and Speed in Two Dimensions Average velocity: Average speed:v = s ΔtΔt ΔrΔr ΔtΔt d ΔtΔt =

shows the velocity of an object at any given moment points in the direction of movement at that instant Instantaneous Velocity Vector

equal to the magnitude of the instantaneous velocity 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 Speed

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 average acceleration vector is equal to the velocity difference divided by the time interval: a = v 2 – v 1 ΔtΔt ΔtΔt ΔvΔv =

Acceleration in Two Dimensions The direction of the average acceleration is always the same direction as the velocity difference vector, Δ v.

Instantaneous Acceleration acceleration at a particular moment Its vector points in the same direction as the instantaneous velocity difference vector.

Projections

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 Trajectory: the path of a projectile

Projectiles Ballistic trajectory: the unpowered portion of a projectile’s path gravitational force only air resistance will be disregarded

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

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

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 Component The horizontal displacement is sometimes called the range. recall the first equation of motion: v 2x = v 1x + a x Δt

Horizontal Component Since the horizontal acceleration is zero, we now have: v 2x = v 1x

Horizontal Component Similarly, the second equation of motion becomes: x 2 = x 1 + v x Δt d x = x 2 - x 1 = v x Δt

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

Vertical Component downward acceleration is g = m/s² For a horizontal projection, the initial vertical velocity (v 1y ) is zero.

Vertical 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 Component Equations of motion: v 2y = g y Δt d y = ½g y (Δt)² d y = v 2y ² 2g y.

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 motion may appear different to different observers

Projection at an Angle very common in the real world horizontal and vertical accelerations the same as with a horizontal projection a x = 0, a y = -g

Projection at an Angle initial vertical velocity is no longer zero components of initial vertical velocity: v 1x = v 1 cos θ v1 v 1y = v 1 sin θ v1

Projection at an Angle These components can be used in the original equations of motion—no need to memorize another set of equations!

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

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

Projectile 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 Motion The trajectory is symmetrical. Vertical speed is the same at corresponding heights (but the direction has changed).

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

Projectile Motion Therefore, it is often good to know the quadratic formula: -b ± b² - 4ac 2a x =

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.