Kinematics in One Dimension Chapter 2

Expectations After this chapter, students will:  distinguish between distance and displacement  distinguish between speed and velocity  understand what acceleration is  recognize situations in which the kinematic equations apply

Expectations After this chapter, students will:  be able to do calculations involving freely-falling objects

Kinematics Mechanics  Kinematics: the descriptive study of motion, without concern for what causes the motion (space and time only)  Dynamics: the study of what causes and changes the motion of objects (force and inertia)

Displacement and Distance Displacement: the vector that extends from the starting point to the ending point of an object’s travel. Distance: the total length of an object’s path (a scalar quantity).

Displacement and Distance An ant crawls along the edge of a meter stick. He starts out at x 0 = 50 cm Then he moves along to x 1 = 75 cm

Displacement and Distance Now he changes his mind and goes toward the left, to x 2 = 20 cm … … and reverses course yet again and goes to x f = 30 cm, where he collapses in weariness.

Displacement and Distance What distance did the ant travel?  from x 0 = 50 cm to x 1 = 75 cm: d 1 = 25 cm  from x 1 = 75 cm to x 2 = 20 cm: d 2 = 55 cm  from x 2 = 20 cm to x f = 30 cm: d 3 = 10 cm total distance = d 1 + d 2 + d 3 = 25 cm + 55 cm + 10 cm = 90 cm

Displacement and Distance What was the ant’s displacement?  x = x f – x 0 = 30 cm – 50 cm = - 20 cm The magnitude of the displacement was 20 cm; the direction was toward – X (or, as we drew it, leftward).

Displacement and Distance Magnitude of displacement:  x = x f – x 0 Direction of displacement: from x 0 to x f SI unit: meter (m)

Speed and Velocity Back to the ant-infested meter stick: but this time, we’ll take a stopwatch with us.

Speed and Velocity Note that speed is a scalar quantity. Dimensions: length/time SI unit: m/s

Speed and Velocity Velocity is a vector: its direction is the same as that of the displacement vector. Magnitude:

Speed and Velocity Now, the second part of the journey …

Speed and Velocity … and the third.

Speed and Velocity Times and positions: Average speed: tx 0.0 s50 cm 5.0 s75 cm 19 s20 cm 22 s30 cm

Speed and Velocity Times and positions: Average velocity: tx 0.0 s50 cm 5.0 s75 cm 19 s20 cm 22 s30 cm

Acceleration We’ve been talking about “average” velocity because velocity may not be constant as motion occurs. If velocity changes, acceleration occurs.

Acceleration Acceleration is the time rate of change of velocity. If an object has a velocity v 0 at time t 0, and a different velocity v at a later time t, its acceleration is: dimensions: length / time 2 SI units: m/s 2

Acceleration Note that acceleration may or may not involve a change in speed. If either the magnitude (speed) or the direction (or both) of the velocity vector change, an acceleration has taken place. In common speech, “acceleration” means that speed is increasing. In physics, acceleration means: speed increases; speed decreases; direction of motion changes; or some combination of these.

Acceleration Consider an elevator car. At the initial time t 0 = 0 s, it is descending with a velocity v 0 = -1.2 m/s. It comes to a stop (v 1 = 0 m/s) at its floor at time t 1 = 1.5 s. The magnitude of its acceleration is

Acceleration Passengers enter, the door closes, and the elevator starts upward, reaching a constant upward velocity of 1.7 m/s in a time of 3.0 s. The acceleration:

Acceleration The elevator slows as it approaches the 86th floor. At t 3 = 177 s, its velocity (v 3 ) is 1.7 m/s, upward; at t 4 = s, it stops (v 4 = 0 m/s). When the acceleration vector points in the opposite direction from the initial velocity vector, speed decreases.

Kinematic Equations If acceleration is constant: From the definition of acceleration:

Kinematic Equations If acceleration is constant, from the definition of average velocity:

Kinematic Equations If acceleration is constant, we can substitute v from the first kinematic equation into the second:

Kinematic Equations If acceleration is constant, we can solve the definition of acceleration for t: and substitute in the second kinematic equation:

Kinematic Equations Why do we insist that acceleration must be constant? We are implicitly assuming that velocity is a linear function of time.

Kinematic Equations: Summary Valid only when acceleration is constant.

Graphical Representations of Motion Object moving at constant velocity

Graphical Representations of Motion Object moving with changing velocity

Graphical Representations of Motion Object moving with constant velocity

Graphical Representations of Motion Object moving with constant acceleration not true if v is not a linear function of t (constant acceleration)

Free Fall ≈ Constant Acceleration Objects at or near Earth’s surface experience a constant acceleration toward the Earth’s center, due to gravitation – if we ignore the resistive effect of the air through which it falls. The magnitude of this acceleration is m/s 2. The symbol of this acceleration is g.

Free Fall ≈ Constant Acceleration In the presence of air, a falling object has an initial downward acceleration g; the acceleration decreases until the object is falling with a constant velocity called its terminal velocity. The acceleration due to gravity, g, does not depend on the mass or the size of the object.

Free Fall and the Kinematic Equations Consider a stone thrown straight upward with an initial velocity v 0. It will ascend with its velocity decreasing until it stops momentarily; then it will accelerate back downward. Let’s calculate the time, t a, of its ascent.

Free Fall and the Kinematic Equations First, we need to establish a (one- dimensional) coordinate system. Our positive x-axis will point straight up. That means the stone’s initial velocity is +v 0, and its acceleration is –g.

Free Fall and the Kinematic Equations In the first kinematic equation, we know everything except t. We know this because the final velocity, v, is zero (the stone stops momentarily at the top of its trajectory). Another way of looking at this is to say we know v = 0 when t = t a. We can substitute these into the first kinematic equation and solve it for t a.

Free Fall and the Kinematic Equations

Now, let’s calculate the height of the ascent. We know that at the maximum height x max, the stone stops momentarily; that is, v = 0 when x = x max.

Free Fall and the Kinematic Equations Substituting v = 0, a = -g, and x = x max into our fourth kinematic equation:

Free Fall and the Kinematic Equations As the stone falls back to the height (x = 0) from which it started, we wonder: how long does the descent take? Will it take the same amount of time that the ascent did? Let’s find out.

Free Fall and the Kinematic Equations For the descent: v 0 = 0, x = -x max, t = t d, and a = -g. Substitute these into the third kinematic equation:

Free Fall and the Kinematic Equations That result doesn’t look much like our result for t a : But, if we substitute our result for x max into our equation for t d :

Free Fall and the Kinematic Equations By now, you probably suspect that the ascent and descent are fully symmetric, and would probably predict that the final velocity of the falling stone is equal in magnitude and opposite in direction to the initial velocity of the ascending stone. We can check this using the first kinematic equation, substituting our calculated t d for t, -g for a, zero for v 0, and v f for v.

Free Fall and the Kinematic Equations The result: Thus, we have demonstrated that the ascent of an object thrown upward, and its descent to the same height, are indeed fully symmetric. If the object returns to a different height … “all bets are off.”