Questions: A car initially traveling eastward turns north by traveling in a circular path at uniform speed as shown in the figure below. The length of the arc ABC is 245 m, and the car completes the turn in 33.0 s. (a) Determine the car's speed. 7.42 m / s (b) What is the magnitude and direction of the acceleration when the car is at point B? magnitude Your response differs from the correct answer by more than 10%. Double check your calculations. m / s^2 direction - counterclockwise from the +x-axis

Solution Steps

To find the car's speed, we use the formula for speed, which is the distance traveled divided by the time taken. The length of the arc \( ABC \) is given as 245 m, and the car completes the turn in 33.0 s.

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{245 \, \text{m}}{33.0 \, \text{s}} \]

\[ \text{Speed} = \frac{245 \, \text{m}}{33.0 \, \text{s}} \approx 7.42 \, \text{m/s} \]

The car is moving in a circular path, so it experiences centripetal acceleration. The formula for centripetal acceleration \( a_c \) is:

\[ a_c = \frac{v^2}{r} \]

where \( v \) is the speed of the car and \( r \) is the radius of the circular path. To find \( r \), we use the relationship between the arc length \( s \), the radius \( r \), and the central angle \( \theta \) in radians:

\[ s = r \theta \]

Given \( s = 245 \, \text{m} \) and the angle \( \theta = 35^\circ \), we first convert \( \theta \) to radians:

\[ \theta = 35^\circ \times \frac{\pi}{180^\circ} \approx 0.611 \, \text{radians} \]

Now, solve for \( r \):

\[ r = \frac{s}{\theta} = \frac{245 \, \text{m}}{0.611 \, \text{radians}} \approx 401 \, \text{m} \]

Using the speed \( v = 7.42 \, \text{m/s} \) and the radius \( r = 401 \, \text{m} \):

\[ a_c = \frac{v^2}{r} = \frac{(7.42 \, \text{m/s})^2}{401 \, \text{m}} \approx 0.137 \, \text{m/s}^2 \]

Final Answer