Aparticle is moving around in a circle and its position is given in polar coordinates as x = rcosθ, and y = rsinθ, where r is the radius of the circle, and θ is in radians. from these equations derive the equation for centripetal acceleration.

“Without loss of generality, we only need to look at the equation for the x-position, since we know that centripetal acceleration points towards the center of the circle. Thus, when θ = 0, the second derivative of x with respect to time must be the centripetal acceleration.

The first derivative of x with respect to time t is:

dx/dt = —Rsinθ(dθ/dt)

The second derivative of x with respect to time t is:

d2x/dt2 = —Rcosθ(dθ/dt)2—Rsinθ(d2θ/dt2)

In both of the above equations the chain rule of Calculus is used and by assumption θ is a function of time. Therefore, θ can be differentiated with respect to time.

Now, evaluate the second derivative at θ = 0.

We have,

d2x/dt2 = —R(dθ/dt)2

The term dθ/dt is usually called the angular velocity, which is the rate of change of the angle θ. It has units of radians/second.

For convenience we can set w ≡ dθ/dt.

Therefore,

d2x/dt2 = —Rw2

This is the well-known form for the centripetal acceleration equation.”

-Leah

i had this question and the answer was 3.15 miles.

martina travels from school to her mom's office. from there, she walks to school. after school, she walks to the library and then to a friends house before finally ending back at home. find martina's total distance traveled.

a) 0 miles  

b) 0.4 miles  

c) 1 mile  

d) 3.15 miles

e/c, where e is its total energy