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## Q. 7.5

AT THE RACETRAK

GOAL Apply the concepts of centripetal acceleration and tangential velocity.

PROBLEM A race car accelerates uniformly from a speed of $40.0 \mathrm{~m} / \mathrm{s}$ to a speed of $60.0 \mathrm{~m} / \mathrm{s}$ in $5.00 \mathrm{~s}$ while traveling counterclockwise around a circular track of radius $4.00 \times 10^2 \mathrm{~m}$. When the car reaches a speed of $50.0 \mathrm{~m} / \mathrm{s}$, calculate (a) the magnitude of the car’s centripetal acceleration, (b) the angular velocity, (c) the magnitude of the tangential acceleration, and (d) the magnitude of the total acceleration.

STRATEGY Substitute values into the definitions of centripetal acceleration (Eq. 7.13),

$a_c=\frac{v^2}{r}$      [7.13]

tangential velocity (Eq. 7.10),

$v_t = rω$      [7.10]

and total acceleration (Eq. 7.18).

$a = \sqrt{a_t^2 + a_c^2}$      [7.18]

Dividing the change in tangential velocity by the time yields the tangential acceleration.

## Verified Solution

(a) Calculate the magnitude of the centripetal acceleration when $v=50.0 \mathrm{~m} / \mathrm{s}$.

Substitute into Equation 7.13:

$a_c=\frac{v^2}{r}=\frac{(50.0 \mathrm{~m} / \mathrm{s})^2}{4.00 \times 10^2 \mathrm{~m}}=6.25 \mathrm{~m} / \mathrm{s}^2$

(b) Calculate the angular velocity.

Solve Equation $7.10$ for $\omega$ and substitute:

$\omega=\frac{v}{r}=\frac{50.0 \mathrm{~m} / \mathrm{s}}{4.00 \times 10^2 \mathrm{~m}}=0.125 \mathrm{rad} / \mathrm{s}$

(c) Calculate the magnitude of the tangential acceleration.

Divide the change in tangential velocity by the time:

$a_t=\frac{v_f-v_i}{\Delta t}=\frac{60.0 \mathrm{~m} / \mathrm{s}-40.0 \mathrm{~m} / \mathrm{s}}{5.00 \mathrm{~s}}=4.00 \mathrm{~m} / \mathrm{s}^2$

(d) Calculate the magnitude of the total acceleration.

Substitute into Equation 7.18:

\begin{aligned}&a=\sqrt{a_t^2+a_c^2}=\sqrt{\left(4.00 \mathrm{~m} / \mathrm{s}^2\right)^2+\left(6.25 \mathrm{~m} / \mathrm{s}^2\right)^2} \\&a=7.42 \mathrm{~m} / \mathrm{s}^2\end{aligned}

REMARKS We can also find the centripetal acceleration by substituting the derived value of $\omega$ into Equation 7.17.

$a_c = \frac{r^2 ω^2}{r} = r ω^2$      [7.17]