Question 11.SP.16: A motorist is traveling on a curved section of highway with ...
A motorist is traveling on a curved section of highway with a radius of 2500 ft at a speed of 60 mi/h. The motorist suddenly applies the brakes,causing the automobile to slow down at a constant rate. If the speed has been reduced to 45 mi/h after 8 s, determine the acceleration of the auto-mobile immediately after the brakes have been applied.
STRATEGY: You know the path of the motion, and that the forward speed of the vehicle defines the direction of e_t. Therefore, you can use tangential and normal components.
MODELING and ANALYSIS:
Tangential Component of Acceleration. First express the speeds in ft/s.
Since the automobile slows down at a constant rate, you have the tangen-
tial acceleration of
Normal Component of Acceleration. Immediately after the brakes have been applied, the speed is still 88 ft/s. Therefore, you have
a_{n}=\frac{v^{2}}{\rho}=\frac{(88 \mathrm{ft} / \mathrm{s})^{2}}{2500 \mathrm{ft}}=3.10 \mathrm{ft} / \mathrm{s}^{2}Magnitude and Direction of Acceleration. The magnitude and direction of the resultant a of the components an and at are (Fig. 1)
\begin{array}{rlr}\tan \alpha & =\frac{a_{n}}{a_{t}}=\frac{3.10 \mathrm{ft} / \mathrm{s}^{2}}{2.75 \mathrm{ft} / \mathrm{s}^{2}} \quad \alpha=48.4^{\circ} \\a & =\frac{a_{n}}{\sin \alpha}=\frac{3.10 \mathrm{ft} / \mathrm{s}^{2}}{\sin 48.4^{\circ}} \quad a=4.14 \mathrm{ft} / \mathrm{s}^{2}\end{array}REFLECT and THINK: The tangential component of acceleration is opposite the direction of motion, and the normal component of acceleration points to the center of curvature, which is what you would expect for slowing down on a curved path. Attempting to do this problem in Cartesian coordinates is quite difficult.
