- #1

Arkun

- 4

- 0

## Homework Statement

A car drives on a circular road with radius ##R##. The distance driven by the car is given by ##d(t) = at^3 + bt## [where ##t## in seconds will give ##d## in meters]. In terms of ##a##, ##b##, and ##R##, and when ##t = 2## seconds, find an expression for the magnitudes of (i) the tangential acceleration ##a_{tan}##, and (ii) the radial acceleration ##a_{rad}##.

## Homework Equations

$$d(t) = at^3 + bt$$

$$a_{rad} = \frac{v^2}{r}$$

$$a_{tan} = \frac{d|\vec v|}{dt}$$

## The Attempt at a Solution

For i) I took the second derivative of ##d(t)## and plugged in ##t = 2## to get:

$$a_{tan} = 6a(2)$$

My reasoning for this is that ##d(t)## describes the distance the car traveled around circular road, which means that its derivative will describe the car's tangential velocity at time ##t##, and ultimately means that its second derivative will be the tangential acceleration.

For ii) I took the first derivative of ##d(t)## because, like with part i), I reasoned it must be the tangential velocity of the car since ##d(t)## describes the distance driven by the car along the curved road. So I first took the derivative of ##d(t)## and got:

$$v(t) = 3a(2)^2 + b$$

Since the formula for ##a_{rad}## is quotient of the square of the tangential velocity and the radius of the circle, I plugged ##v(t)## into ##v## and ##R## into ##r## for the formula of ##a_{rad}## to get:

$$a_{rad} = \frac{(3a(2)^2 + b)^2}{R}$$

The issue I'm having with this problem is that I don't meet the conditions stated in the problem for part i): I don't have ##a_{tan}## in terms of ##a##,##b##, and ##R##. So what I'm left wondering is whether ##v(t)## actually is the tangential velocity or if I need to use another formula that will help me meet the conditions stated in the problem. Since ##a(t)## is not zero I'm assuming the car isn't moving at uniform circular motion, so uniform circular motion equations don't apply.