Question 5.6: The Runaway Car A car of mass m is on an icy driveway inclin...

(A) Conceptualize Use Figure 5.12a to conceptualize the situation. From everyday experience, we know that a car on an icy incline will accelerate down the incline. (The same thing happens to a car on a hill with its brakes not set.)

Categorize We categorize the car as a particle under a net force because it accelerates. Furthermore, this example belongs to a very common category of problems in which an object moves under the influence of gravity on an inclined plane.

Analyze Figure 5.12b shows the free-body diagram for the car. The only forces acting on the car are the normal force \overrightarrow{n} exerted by the inclined plane, which acts perpendicular to the plane, and the gravitational force \overrightarrow{F}_g=m\overrightarrow{g}, which acts vertically downward. For problems involving inclined planes, it is convenient to choose the coordinate axes with x along the incline and y perpendicular to it as in Figure 5.12b. Using similar triangles, we can show that the angle between the gravitational force \overrightarrow{F}_g and the negative y axis in part b of Figure 5.12 is equal to the angle θ that the incline makes with the horizontal in part a. With these axes, we represent the gravitational force by a component of magnitude mg sin θ along the positive x axis and one of magnitude mg cos θ along the negative y axis. Our choice of axes results in the car being modeled as a particle under a net force in the x direction and a particle in equilibrium in the y direction.

Apply these models to the car:

(1)   \sum F_x=m g \sin \theta=m a_x

(2)   \sum F_y=n-m g \cos \theta=0

Solve Equation (1) for a_x:

(3)   a_x=g \sin \theta

Finalize Note that the acceleration component a_x is independent of the mass of the car. It depends only on the angle of inclination and on g.

From Equation (2), we conclude that the component of \overrightarrow{F}_g perpendicular to the incline is balanced by the normal force; that is, n = mg cos θ. This situation is a case in which the normal force is not equal in magnitude to the weight of the object (as discussed in Pitfall Prevention 5.6 on page 104).

It is possible, although inconvenient, to solve the problem with “standard” horizontal and vertical axes. You may want to try it, just for practice.

(B) Conceptualize Imagine the car is sliding down the hill and you use a stopwatch to measure the entire time interval until it reaches the bottom.

Categorize This part of the problem belongs to kinematics rather than to dynamics, and Equation (3) shows that the acceleration a_x is constant. Therefore, you should categorize the car in this part of the problem as a particle under constant acceleration.

Analyze Defining the initial position of the front bumper as x_i = 0 and its final position as x_f = d, and recognizing that v_{xi} = 0, choose Equation 2.16 from the particle under constant acceleration model:

x_f=x_i+v_{x i} t+\frac{1}{2} a_x t^2 \quad \text { (for constant } a_x)     (2.16)

Solve for t:

(4)   t=\sqrt{\frac{2 d}{a_x}}=\sqrt{\frac{2 d}{g \sin \theta}}

Use Equation 2.17, with v_{xi} = 0, to find the final velocity of the car:

v_{x f}{}^2=v_{x i}{ }^2+2 a_x\left(x_f-x_i\right) \quad \text { (for constant } a_x)     (2.17)

(5)   v_{x f}=\sqrt{2 a_x d}=\sqrt{2 g d \sin \theta}

Finalize We see from Equations (4) and (5) that the time t at which the car reaches the bottom and its final speed v_{xf} are independent of the car’s mass, as was its acceleration. Notice that we have combined techniques from Chapter 2 with new techniques from this chapter in this example. As we learn more techniques in later chapters, this process of combining analysis models and information from several parts of the book will occur more often. In these cases, use the Analysis Model Approach to Problem Solving discussed in Chapter 2 to help you work your way through new problems.