Chapter 5.4, Problem 1IE

When the speed of the race car in Example 5–16 is 30 m/s, how are (a) a_tan and (b) a_R changed?

EXAMPLE 5–16 Two components of acceleration. A race car starts from rest in the pit area and accelerates at a uniform rate to a speed of 35 m/s in 11 s, moving on a circular track of radius 500 m. Assuming constant tangential acceleration, find (a) the tangential acceleration, and (b) the radial acceleration, at the instant when the speed is v = 15 m/s.

APPROACH The tangential acceleration relates to the change in speed of the car, and can be calculated as a_tan = Δv/Δt. The centripetal acceleration relates to the change in the direction of the velocity vector and is calculated using a_R = v²/r.

SOLUTION (a) During the 11-s time interval, we assume the tangential acceleration a_tan is constant. Its magnitude is

$a_{\tan }=\frac{\Delta v}{\Delta t}=\frac{\left(35\text{m}/\text{s}-0\text{m}/\text{s}\right)}{11\text{s}}=3.2{\text{m}/\text{s}}^2.$

(b) When $v=15\text{m}/\text{s}$, the centripetal acceleration is

$a_{\text{R}}=\frac{v^2}{r}=\frac{{\left(15\text{m}/\text{s}\right)}^2}{\left(500\text{m}\right)}=0.45{\text{m}/\text{s}}^2.$

NOTE The radial acceleration increases continually, whereas the tangential acceleration stays constant.