1.4.1 Motion Diagrams with Velocity Vectors

The velocity vector points in the same direction as the displacement Δr⃗, and the length of v ⃗ is directly proportional to the length of Δr⃗. Consequently, the vectors connecting each dot of a motion diagram to the next, which we previously labeled as displacements, could equally well be identified as velocity vectors.

This idea is illustrated in FIGURE 1.10, which shows four frames from the motion diagram of a tortoise racing a hare. The vectors connecting the dots are now labeled as velocity vectors v ⃗. The length of a velocity vector represents the average speed with which the object moves between the two points. Longer velocity vectors indicate faster motion. You can see that the hare moves faster than the tortoise.

FIGURE 1.10 Motion diagram of the tortoise racing the hare.

The figure illustrates the motion diagram of the tortoise racing the hare.
FIGURE 1.10 Full Alternative Text

Notice that the hare’s velocity vectors do not change; each has the same length and direction. We say the hare is moving with constant velocity. The tortoise is also moving with its own constant velocity.

EXAMPLE 1.2 Accelerating up a hill

The light turns green and a car accelerates, starting from rest, up a 20° hill. Draw a motion diagram showing the car’s velocity.

MODEL Use the particle model to represent the car as a dot.

VISUALIZE The car’s motion takes place along a straight line, but the line is neither horizontal nor vertical. A motion diagram should show the object moving with the correct orientation—in this case, at an angle of 20°. FIGURE 1.11 shows several frames of the motion diagram, where we see the car speeding up. The car starts from rest, so the first arrow is drawn as short as possible and the first dot is labeled “Start.” The displacement vectors have been drawn from each dot to the next, but then they are identified and labeled as average velocity vectors v ⃗.

FIGURE 1.11 Motion diagram of a car accelerating up a hill.

The figure illustrates the motion diagram of a car accelerating up a hill.

EXAMPLE 1.3 A rolling soccer ball

Marcos kicks a soccer ball. It rolls along the ground until stopped by Jose. Draw a motion diagram of the ball.

MODEL This example is typical of how many problems in science and engineering are worded. The problem does not give a clear statement of where the motion begins or ends. Are we interested in the motion of the ball just during the time it is rolling between Marcos and Jose? What about the motion as Marcos kicks it (ball rapidly speeding up) or as Jose stops it (ball rapidly slowing down)? The point is that you will often be called on to make a reasonable interpretation of a problem statement. In this problem, the details of kicking and stopping the ball are complex. The motion of the ball across the ground is easier to describe, and it’s a motion you might expect to learn about in a physics class. So our interpretation is that the motion diagram should start as the ball leaves Marcos’s foot (ball already moving) and should end the instant it touches Jose’s foot (ball still moving). In between, the ball will slow down a little. We will model the ball as a particle.

VISUALIZE With this interpretation in mind, FIGURE 1.12 shows the motion diagram of the ball. Notice how, in contrast to the car of Figure 1.11, the ball is already moving as the motion diagram video begins. As before, the average velocity vectors are found by connecting the dots. You can see that the average velocity vectors get shorter as the ball slows. Each v  ⃗ is different, so this is not constant-velocity motion.

FIGURE 1.12 Motion diagram of a soccer ball rolling from Marcos to Jose.

The figure illustrates the motion diagram of a soccer ball rolling from Marcos to Jose.