AP Physics ST Displacement, Velocity, Acceleration Vectors Projectile Motion

Displacement, Velocity, Acceleration Vectors True or False?? The characteristics of motion of a moving object are completely known IF the position as a function of time is known!

Displacement, Velocity, Acceleration Vectors TRUE!! “Characteristics of motion”… The most commonly asked questions regarding the motion of an object are… – HOW FAST? – HOW FAR? Let’s consider this idea from a vector point of view! bay.scdsb.on.ca

Displacement, Velocity, Acceleration Vectors Consider an object moving along the path described. r i and r f describe position vectors relative to a fixed frame of reference. y x path of object riri rfrf

Displacement, Velocity, Acceleration Vectors Δr represents the displacement vector… – straight-line distance between two points. The “rate” of change of displacement describes velocity… y x path of object riri rfrf ΔrΔr

Displacement, Velocity, Acceleration Vectors Average velocity is path independent – Depends only on initial and final locations. Instantaneous velocity occurs when the time span between the two positions becomes increasingly smaller… – In essence, instantaneous velocity is the limit of the average velocity.

Displacement, Velocity, Acceleration Vectors Consider an object with velocities v i and v f at locations r i and r f. The “rate” at which the object changes its velocity describes its acceleration… riri rfrf vivi vfvf y x

Displacement, Velocity, Acceleration Vectors Δv represents the change in the velocity vector. Acceleration occurs when the vector Δv… 1.changes in magnitude. 2.changes in direction. 3.changes in both magnitude AND direction. riri rfrf -v i vfvf y x Δv vfvf vivi

Displacement, Velocity, Acceleration Vectors Average acceleration is path independent – Depends only on initial and final instantaneous velocities. Instantaneous acceleration occurs when the time span between the two instantaneous velocities becomes increasingly smaller… – In essence, instantaneous acceleration is the limit of the average acceleration.

Displacement, Velocity, Acceleration Vectors Recall the initial premise… – The characteristics of motion of a moving object are completely known IF the position as a function of time is known! So, from vectors can HOW FAST the object is moving and HOW FAR it will travel be determined? – Bring it all together! r rxrx ryry vxvx vyvy v y x

Displacement, Velocity, Acceleration Vectors The position of the object can be described as… Knowing the position vector the velocity can be determined… r rxrx ryry vxvx vyvy v y x

Displacement, Velocity, Acceleration Vectors Assuming constant acceleration, a x and a y remain constant. Thus we can apply the kinematic expressions to determine HOW FAST and HOW FAR. r rxrx ryry vxvx vyvy v y x

Displacement, Velocity, Acceleration Vectors Note that HOW FAST applies to both the x and the y directions… Substituting into resultant form…

Displacement, Velocity, Acceleration Vectors resultant velocity constant acceleration

Displacement, Velocity, Acceleration Vectors Known as HOW FAST… Meaning…the final velocity of an object depends on two factors 1.how fast the object was moving initially, plus 2.any velocity acquired resulting from constant acceleration.

Displacement, Velocity, Acceleration Vectors Similarly HOW FAR applies to both the x and the y directions… Substituting into resultant form…

Displacement, Velocity, Acceleration Vectors resultant initial position displacement resulting form initial velocity displacement resulting from constant acceleration

Displacement, Velocity, Acceleration Vectors Known as HOW FAR… Meaning…the total displacement of an object depends on three factors: 1.the initial position relative to the frame of reference, plus 2.how far the object traveled while moving initially, plus 3.any displacement acquired resulting from constant acceleration.

Projectile Motion Projectile Motion – – Any object traveling with no means of self propulsion – under the influence of gravity alone. Trajectory – – Path that a projectile follows. Parabola – – Shape of the trajectory.

Projectile Motion Assumptions: 1.Free-fall acceleration g is constant in magnitude (9.8m/s 2 ) and directed down 2.Air resistance is negligible. web.mst.edu

Projectile Motion Projectile motion is the superposition of two motions 1.Constant velocity in the horizontal direction (b/c there is no acceleration; air resistance is negligible), and 2.Free-fall motion in the vertical direction. lnk2lrn.com

Projectile Motion This understanding enables one to consider the 1.Horizontal component of motion 2.Vertical component of motion lnk2lrn.com

Projectile Motion 1.Initial horizontal component of motion: lnk2lrn.com

Projectile Motion 2.Initial vertical component of motion: lnk2lrn.com

Projectile Motion “air resistance is negligible” means… Resolving motion into horizontal and vertical components enables one to consider HOW FAST and HOW FAR…

Projectile Motion Horizontal Motion: HOW FAST at any instant along the trajectory… HOW FAR (position) at any instant along the trajectory…

Projectile Motion Vertical Motion: HOW FAST at any instant along the trajectory… HOW FAR (position) at any instant along the trajectory…

Projectile Motion To reaffirm the concept that … “Projectile motion is the superposition of two motions: 1.constant velocity in the horizontal direction, and 2.free-fall motion in the vertical direction.” Consider the following diagram…

Projectile Motion mathworks.com NO-GRAVITY LINE

Projectile Motion IF there were no gravity, the object would continue to travel along the straight-line path covering a distance defined by… mathworks.com NO-GRAVITY LINE

Projectile Motion mathworks.com NO-GRAVITY LINE The vertical distance through which the particle “falls off” the straight-line path is the same distance that a freely falling body would fall if dropped from rest in the same time interval.

Projectile Motion Max Height, Max Range The highest point that a projectile can reach and the furthest point a projectile can reach depend on two factors: 1.Initial velocity and initial launch angle, and 2.ONLY IF the projectile lands at the same height from which it was launched. galileoandeinstein.physics.virginia.edu

Projectile Motion Max Height, Max Range Max Height (H): – The vertical velocity at max height = 0 m/s. – Time to peak: – Max height (H)… ngsir.netfirms.com

Projectile Motion Max Height, Max Range Max Range (R): – Recall time to peak:... and double it! – Max Range (R)… onlinephys.com

Lesson Summary

Example #1 AP-C Mech 85-3ab

Example #2 AP-B Mech 86-2ab

Example #3 AP-B Mech 94-1bc