Kinematics of Two-Dimensional Motion Position Vectors • 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 Frame of Reference • The coordinate system within which motion is measured or observed • There is no absolute frame of reference. Displacement Change in position: Δr d = Δr = r2 – r1 r2 is the position at the end r1 is the initial position Displacement Displacement is the same regardless of the reference frame used! Velocity and Speed in Two Dimensions Δr d Average velocity: v = Δt = Δt Average speed: s v= Δt Instantaneous Velocity Vector • shows the velocity of an object at any given moment • points in the direction of movement at that instant Instantaneous Speed • equal to the magnitude of the instantaneous velocity v = |v| Average Speed • is often quite different from the magnitude of the average velocity • Average speed equals average velocity only when s = |d|. 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: v2 – v 1 Δv a= = Δt Δt Acceleration in Two Dimensions The direction of the average acceleration is always the same direction as the velocity difference vector, Δv. 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: v2x = v1x + axΔt Horizontal Component • Since the horizontal acceleration is zero, we now have: v2x = v1x Horizontal Component • Similarly, the second equation of motion becomes: x2 = x1 + vxΔt dx = x2 - x1 = vxΔt Horizontal Component • The third equation of motion becomes meaningless since it has a denominator of zero. Vertical Component • downward acceleration is g = -9.81 m/s² • For a horizontal projection, the initial vertical velocity (v1y) 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: v2y = gyΔt dy = ½gy(Δt)² v2y² dy = 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 • ax = 0, ay = -g 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 • 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 x= 2a 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.
