The race track shown in Fig. 6-81 has two straight sections...

The race track shown in Fig. $6-81$ has two straight sections connected by semicircular ends. A car is traveling in a clockwise direction around the track at a constant speed. Assume that air resistance is negligible. Draw three sketches of the race track. (a) On the first sketch show the velocity vector at each of the numbered points $1-4 .$ Make the relative lengths of the vectors consistent with the relative magnitudes of the velocity at the four points. (b) On the second sketch show the acceleration vectors at each of the numbered points $1-4 .$ Make the relative lengths of the vectors consistent with the relative magnitudes of the acceleration at the four points. Hint: Use the techniques developed in Chapter 5 to draw vectors representing the acceleration or change in velocity. (c) Horizontal forces are needed to maintain the car's motion around the track. These are provided by road friction and by road forces where the track is banked at the curves. On the third sketch show the vectors representing the required horizontal forces at each of the numbered points $1-4$. Make the relative lengths of the vectors consistent with the relative magnitudes of the force at the four points.

The race track shown in Fig. $6-81$ has two straight sections connected by semicircular ends. A car is traveling in a clockwise direction around the track at a constant speed. Assume that air resistance is negligible. Draw three sketches of the race track.
(a) On the first sketch show the velocity vector at each of the numbered points $1-4 .$ Make the relative lengths of the vectors consistent with the relative magnitudes of the velocity at the four points.
(b) On the second sketch show the acceleration vectors at each of the numbered points $1-4 .$ Make the relative lengths of the vectors consistent with the relative magnitudes of the acceleration at the four points. Hint: Use the techniques developed in Chapter 5 to draw vectors representing the acceleration or change in velocity.
(c) Horizontal forces are needed to maintain the car's motion around the track. These are provided by road friction and by road forces where the track is banked at the curves. On the third sketch show the vectors representing the required horizontal forces at each of the numbered points $1-4$. Make the relative lengths of the vectors consistent with the relative magnitudes of the force at the four points.

Key Concepts

Centripetal (Radial) Acceleration

This concept explains that for an object moving along a curved or circular path, there is an acceleration directed toward the center of curvature, known as centripetal acceleration. For the car on the track, even at constant speed, this inward-pointing acceleration is essential to change the car’s direction along the curved portions and is represented by vectors pointing towards the center of the track’s curvature.

Newton’s Second Law and Horizontal Forces

Newton’s Second Law connects the net force acting on an object with its acceleration. In the context of the race track, the necessary horizontal forces (provided by friction and banking) counteract the inertial tendency of the car to move in a straight line. These forces are responsible for providing the centripetal acceleration that keeps the car on the curved path, and their representations must mirror the relative magnitudes and directions of the corresponding acceleration vectors.

Uniform Motion and Constant Speed

This concept emphasizes that when an object moves with a constant speed, its acceleration is solely due to changes in the direction of its motion rather than changes in the magnitude of its velocity. In the context of the race track, although the car’s speed remains constant, its direction continuously changes especially along the curved (semicircular) sections, introducing a non-zero acceleration component.

Velocity as a Tangential Vector

In any curved motion, the velocity vector at any point is always tangent to the path. For the race track, this means that at each numbered point, the velocity vector’s direction aligns with the instant direction of the car's travel along the track. This concept is crucial for sketching accurate representations of the car’s velocity throughout its motion.

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