# CA Model Concepts

Constant Acceleration Model Concepts

By Marc Reif

The paradigm for this unit is the ball bearing rolling down a ramp. As the ball bearing travels down the ramp, it gains more position in equal intervals of time (it speeds up). We found that the velocity of the ball bearing changed by equal amounts in equal time intervals. Upon examination of the motion of the ball bearing, we defined it as constant acceleration.

Average Acceleration (a bar) is defined as the change in velocity over the change in time. It is a vector and is the slope of the v-t graph. A shorthand version of this definition is a = v/t.

Acceleration is a rate of change of a rate of change. An object that accelerates at 9.8 m/s2 or m/s/s gains 9.8 m/s of velocity for every second it accelerates.

If the pos-t graph is a curve, the object is accelerating. Curves that we see (to describe motion in this class) usually fit a power regression (to the second power) or a quadratic regression. Each term in the regression equation has some physical meaning. The shape of these curves is parabolic. The form of the model for an object undergoing constant acceleration is y=Ax2+Bx+C.

If the pos-t graph is a curve, the slope of a tangent line to the curve is the instantaneous velocity at that point on the graph (the velocity at that moment).

The intercept of the pos-t graph is once again the initial position.

When an object undergoing constant acceleration changes direction, its instantaneous velocity is zero. However, its acceleration is not zero.

If the v-t graph is a curve, the acceleration is changing. The slope of the tangent line is the instantaneous acceleration for this graph.

A positive acceleration makes something traveling in the positive direction speed up, and an object traveling in the negative direction slows down. A negative acceleration makes something traveling in the negative direction speed up, and an object traveling in the positive direction slows down.

Deceleration is when an object is slowing down (speed is going toward zero). It is not the same thing as a negative acceleration (which can represent speeding up in a negative direction).

Working from the definitions above, we can derive useful equations that describe the motion of objects.

In a vacuum, all falling objects accelerate at the same rate. The shorthand for this rate is g (g on the earth = 9.8 m/s2).

Objects accelerate at the same rate on earth b/c even though more massive objects experience a greater pull of earth's gravity (or weight), their mass (or inertia) resists that pull. Air resistance interferes with this fact in the atmosphere.

The effect of air resistance is more important as objects have the following characteristics: lighter, more surface area, less aerodynamic shape, moving faster. We will usually ignore air resistance b/c it makes the math easier, but for more accurate results, it must be included in many cases.

An object undergoes freefall when only the force of earth's gravity is acting on it in a significant way. Anything thrown up or down, or dropped qualifies as freefalling from the instant it is released.

The instantaneous velocity of an object tossed up is zero at the top of its path. Its acceleration is g.

The speed at equal altitudes of an object tossed up is equal. The velocities are opposites (one is traveling up, the other is down).