To determine the radius of the loop in which the rollercoaster is traveling, we need to relate the given quantities using the formula for centripetal acceleration. The formula for centripetal acceleration [tex]\( a \)[/tex] is given by:
[tex]\[ a = \frac{v^2}{r} \][/tex]
where:
- [tex]\( a \)[/tex] is the centripetal acceleration,
- [tex]\( v \)[/tex] is the velocity, and
- [tex]\( r \)[/tex] is the radius of the loop.
We are given:
- Velocity, [tex]\( v = 16.3 \, \text{m/s} \)[/tex]
- Centripetal acceleration, [tex]\( a = 12.4 \, \text{m/s}^2 \)[/tex]
To find the radius [tex]\( r \)[/tex], we rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{v^2}{a}
\][/tex]
Now substitute the given values into the equation:
[tex]\[
r = \frac{(16.3)^2}{12.4}
\][/tex]
Carrying out the calculations, first compute [tex]\( 16.3^2 \)[/tex]:
[tex]\[
16.3^2 = 265.69
\][/tex]
Next, divide this result by 12.4:
[tex]\[
r = \frac{265.69}{12.4} \approx 21.426612903225806
\][/tex]
Thus, the radius of the loop is:
[tex]\[
r \approx 21.43 \, \text{m}
\][/tex]
