Question 5.16: Two components of acceleration. A race car starts from rest ......
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 \mathrm{~m} / \mathrm{s} in 11 \mathrm{~s}, moving on a circular track of radius 500 \mathrm{~m}. Assuming constant tangential acceleration, find (a) the tangential acceleration, and (b) the radial acceleration, at the instant when the speed is v=15 \mathrm{~m} / \mathrm{s}.
APPROACH The tangential acceleration relates to the change in speed of the car, and can be calculated as a_{\tan }=\Delta v / \Delta t. The centripetal acceleration relates to the change in the direction of the velocity vector and is calculated using a_{\mathrm{R}}=v^{2} / r.
(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{(35 \mathrm{~m} / \mathrm{s}-0 \mathrm{~m} / \mathrm{s})}{11 \mathrm{~s}}=3.2 \mathrm{~m} / \mathrm{s}^{2} .
(b) When v=15 \mathrm{~m} / \mathrm{s}, the centripetal acceleration is
a_{\mathrm{R}}=\frac{v^{2}}{r}=\frac{(15 \mathrm{~m} / \mathrm{s})^{2}}{(500 \mathrm{~m})}=0.45 \mathrm{~m} / \mathrm{s}^{2} .
NOTE The radial acceleration increases continually, whereas the tangential acceleration stays constant.