Figure 6-77 shows a multiple exposure strobe photograph of a...

Figure $6-77$ shows a multiple exposure strobe photograph of a ball rolling on a horizontal table. The image marked with a heavy arrow occurs at time $t=0$ and the ball moves to the right at that instant. Each image of the ball occurs $1 / 30 \mathrm{~s}$ later than the one immediately to its left. Using the coordinate system shown in Fig. 6-77, sketch qualitatively accurate (i.e., we don't care about the values but we do care about the shape) graphs of each of the following variables as a function of time: $x$ coordinate, $y$ coordinate, $x$ -component of velocity, $y$ -component of velocity, $x$ -component of the net force on the ball, and $y$ -component of the net force on the ball. The time at which the "kink" in the path occurs is $t=t_{1} .$ Be sure to note this important time on your graphs.

Key Concepts

Impulsive Forces and Sudden Changes in Motion

When an object experiences a sudden force over a short time interval, its velocity can change abruptly, which is often represented as a distinct 'kink' in a velocity-time graph. Understanding how impulses affect the momentum of an object is critical for interpreting moments of sudden motion alteration and the resulting shifts in the corresponding force and acceleration graphs.

Newton’s Second Law of Motion

Newton’s second law, which states that force equals mass times acceleration, underlies the connection between the net force acting on an object and the resulting change in velocity. This fundamental principle informs how acceleration manifests as a change in the slope of the velocity graphs, and any sudden change in force (such as an impulsive force) produces a corresponding sudden change in velocity.

Graphical Interpretation of Motion Variables

Qualitatively sketching graphs for displacement, velocity, and force involves recognizing the typical shapes of these graphs given particular conditions. For instance, a straight-line graph indicates constant velocity, while a curved graph can indicate acceleration. Noting changes or 'kinks' in these graphs is crucial as they signal sudden changes in motion due to abrupt forces.

Independent Component Analysis

In two-dimensional motion, the x and y directions are analyzed separately, as the motion in one direction does not generally affect the motion in the perpendicular direction. This decomposition allows one to describe the displacement, velocity, and acceleration in each independent coordinate separately, leading to a clearer understanding of the overall motion and how forces may act differently along different directions.

Kinematics of Motion

This concept involves describing how an object moves, including the relationships among displacement, velocity, and acceleration. It emphasizes that the motion of an object can be represented graphically over time, with positions and speeds depicted by curves that indicate constant motion or acceleration. In problems involving sketches of the x and y coordinates as well as their derivatives, understanding kinematics is essential to qualitatively capture the behavior of the moving object.

Transcript

00:01 All right.

00:01 So in this problem, we're given this graph here, an x, y, coordinate plane.

00:05 So this is the table.

00:07 And this ball is moving on this table.

00:11 And we need to sketch some graphs that show different variables with respect to time.

00:19 So the time is going to be on our x -axis.

00:22 And so we're going to start with x, the x coordinate.

00:26 And if you look at this graph, if you just focus on the x coordinate, the x -value.

00:31 The x -value, is increasing in a linear fashion.

00:35 As we move along in time, the x variable is increasing with respect to time.

00:42 And since this is t0, our x value is increasing with respect to time in a linear fashion.

00:51 And t1 happens six seconds in or six 30th of a second in.

00:56 So let's just call that t1 right there.

00:59 So it's not really very interesting in the x direction.

01:03 For the y coordinate now we're looking at this the y coordinate it's stuck at a single position in the y coordinate for the first six marks and then it begins to increase so our y value doesn't change with respect to time until we reach that t1 and then at that t1 y begins to increase in a linear fashion so y is going to actually look very similar to the actual graph that we're starting from.

01:41 The next graph we need to make is going to be the x component of the velocity.

01:49 So this is the sub x or the x component of the velocity.

01:54 And again, this is moving at a constant speed.

01:57 So our x component of the velocity, it's just going to be a constant value...

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