A car of mass m is on an icy driveway inclined at an angle θ, as in Figure 5.11a. (A) Find the acceleration of the car, assuming that the driveway is frictionless.

Question

A car of mass m is on an icy driveway inclined at an angle θ, as in Figure 5.11a.

(A) Find the acceleration of the car, assuming that the driveway is frictionless.

(B) Suppose the car is released from rest at the top of the incline, and the distance from the car’s front bumper to the bottom of the incline is d. How long does it take the front bumper to reach the bottom, and what is the car’s speed as it arrives there?

Accepted Answer

Conceptualize the situation using Figure 5.11a. From everyday experience, we know that a car on an icy incline will accelerate down the incline. (It will do the same thing as a car on a hill with its brakes not set.) This allows us to categorize the situation as a nonequilibrium problem—that is, one in which an object accelerates. Figure 5.11b shows the free-body diagram for the car, which we can use to analyze the problem.
The only forces acting on the car are the normal force n exerted by the inclined plane, which acts perpendicular to the plane, and the gravitational force [latex] F _{g}=m g [/latex], 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 5.11b. (It is possible to solve the problem with “standard” horizontal and vertical axes. You may want to try this, just for practice.) Then, we replace the gravitational force by a component of magnitude [latex] m g \sin heta [/latex] along the positive x axis and one of magnitude [latex] m g \cos heta [/latex] along the negative y axis.
Now we apply Newton’s second law in component form, noting that [latex] a_{y}=0 [/latex]:

[latex] ext { (1) } \quad \sum F_{x}=m g \sin heta=m a_{x} [/latex].

[latex] ext { (2) } \quad \sum F_{y}=n-m g \cos heta=0 [/latex].

Solving (1) for [latex] a_{x} [/latex] , we see that the acceleration along the incline is caused by the component of [latex] F _{g} [/latex] directed down the incline:

(3) [latex] a_{x}=g \sin heta [/latex].

To finalize this part, note that this acceleration component is independent of the mass of the car! It depends only on the angle of inclination and on g.
From (2) we conclude that the component of [latex] F _{g} [/latex] perpendicular to the incline is balanced by the normal force; that is, [latex] n=m g \cos heta [/latex]. This is another example of a situation in which the normal force is not equal in magnitude to the weight of the object.

(B) Suppose the car is released from rest at the top of the incline, and the distance from the car’s front bumper to the bottom of the incline is d. How long does it take the front bumper to reach the bottom, and what is the car’s speed as it arrives there?

Conceptualize by imagining that the car is sliding down the hill and you are operating a stop watch to measure the entire time interval until it reaches the bottom. This part of the problem belongs to kinematics rather than to dynamics, and Equation (3) shows that the acceleration [latex] a_{x} [/latex] is constant. Therefore you should categorize this problem as that of a particle undergoing constant acceleration. Apply Equation 2.12, [latex] x_{f}=x_{i}+v_{x i} t+\frac{1}{2} a_{x} t^{2} [/latex] to analyze the car’s motion. Defining the initial position of the front bumper as [latex] x_{i}=0 ext { and its final position as } x_{f}=d[/latex], and recognizing that [latex] v_{x i}=0 [/latex], we obtain

[latex] x_{f}=x_{i}+v_{x i} t+\frac{1}{2} a_{x} t^{2} \quad ext { (for constant } a_{x} ext { ) } [/latex] (2.12).

[latex] d=\frac{1}{2} a_{x} t^{2} [/latex].

(4) [latex] t=\sqrt{\frac{2 d}{a_{x}}}=\sqrt{\frac{2 d}{g \sin heta}} [/latex].

[latex] ext { Using Equation 2.13, with } v_{x i}=0 ext {, we find that } [/latex].

[latex] v_{x f}^{2}=v_{x i}{ }^{2}+2 a_{x}\left(x_{f}-x_{i} ight) \quad ext { (for constant } a_{x} ext { ) } [/latex] (2.13).

[latex] v_{x f}^{2}=2 a_{x} d [/latex].

(5) [latex] v_{x f}=\sqrt{2 a_{x} d}=\sqrt{2 g d \sin heta} [/latex].

To finalize this part of the problem, we see from Equations (4) and (5) that the time t at which the car reaches the bottom and its final speed [latex] v_{x f} [/latex] are independent of the car’s mass, as was its acceleration. Note that we have combined techniques from Chapter 2 with new techniques from the present chapter in this example. As we learn more and more techniques in later chapters, this process of combining information from several parts of the book will occur more often. In these cases, use the General Problem- Solving Strategy to help you identify what techniques you will need.

Question 5.6: A car of mass m is on an icy driveway inclined at an angle θ...

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