# 1 D Kinematics Particle Under Constant Acceleration Average

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1 -D Kinematics Particle Under Constant Acceleration

Average Acceleration l Acceleration is the rate of change of the velocity: l SI units are m/s² In one dimension, positive and negative can be used to indicate direction. l

Instantaneous Acceleration l The instantaneous acceleration is the limit of the average acceleration as t approaches 0 l The term acceleration will mean instantaneous acceleration l If average acceleration is wanted, the word average will be included

Acceleration: velocity – time graph l The slope on the velocitytime graph reveals the acceleration: l l l Slope of the green (tangent) line represents the instantaneous acceleration. Slope of the blue (secant) line is the average acceleration. Constant acceleration: graph is linear, average and instantaneous values are equal The area under the graph corresponds to the displacement of the particle

Acceleration: position – time graph l Graph curves towards time axis = particle slows down l Graph curves away from time axis = particle slows down Graph is linear = particle moves at constant velocity, so acceleration is zero. l

Acceleration and Velocity l When an object’s velocity and acceleration are in the same direction, the object is speeding up. l When an object’s velocity and acceleration are in opposite directions, the object is slowing down. l If possible, take initial velocity direction as positive. If not, don’t.

Acceleration and Velocity l l Images are equally spaced. The car is moving with constant positive velocity (shown by red arrows maintaining the same size) Acceleration equals zero

Acceleration and Velocity l l Velocity and acceleration are in the same direction Acceleration is uniform (violet arrows, same length) Velocity is increasing (red arrows are getting longer) This shows positive acceleration and positive velocity

Acceleration and Velocity l l Acceleration and velocity are in opposite directions Acceleration is uniform Velocity is decreasing (red arrows are getting shorter) Positive velocity and negative acceleration

Acceleration and Velocity l In all the previous cases, the acceleration was constant l l Shown by the violet arrows all maintaining the same length The diagrams represent motion of a particle under constant acceleration.

Kinematic Equations – The Big Four

Kinematic Equations l The kinematic equations can be used with any particle under uniform acceleration. l You may need to use two (or more) of the equations to solve one problem. l l x, v, a can be either positive or negative! Many times there is more than one correct way to solve a problem.

Kinematic Equations l For constant acceleration: l Can determine an object’s velocity at any time t when we know its initial velocity and its acceleration. l l Assumes ti = 0 and tf = t Does not give any information about displacement.

Kinematic Equations, specific l For constant acceleration: l The average velocity can be expressed as the arithmetic mean of the initial and final velocities.

Kinematic Equations, specific l For constant acceleration: l This gives you the position of the particle in terms of time and velocities. l Doesn’t give you the acceleration.

Kinematic Equations, specific l For constant acceleration: l Gives final position in terms of velocity and acceleration. l Doesn’t tell you about final velocity.

Kinematic Equations, specific l For constant acceleration: l Gives final velocity in terms of acceleration and displacement. l Does not give any information about the time.

When a = 0 l When the acceleration is zero, l l l vxf = vxi = vx xf = xi + vx t i. e. , constant acceleration includes constant velocity as a special case.

Example l A train starts from rest and moves with constant acceleration. It reaches 30 m/s the moment it has traveled 150 m. Calculate: l The acceleration of the train, and the time it takes to travel the first 150 m; l The time it takes, and the distance it travels between reaching 30 m/s until it reaches 50 m/s.