Finding the Velocity and Acceleration of a Car in Circular Motion

[mrcheeses]

Homework Statement

A car enters the circular road with radius r = 200m at a
speed of v = 80km/h. A radar gun at O needs to rotate with constant angular
acceleration d^2θ/dt^2 = 0.025 rad/s2 to follow the motion of the car along the circular road.
a) Determine the acceleration of the car at the instant right after it enters the circular
road;
b) Determine the velocity and acceleration of the car at the instant when the radar
gun has rotated

Homework Equations

v_r = dr/dt
v_θ = r(dθ/dt)
a_r = dr/dt - r(dθ/dt)^2
a_θ = r(d^2θ/dt^2) + 2(dr/dt)θ

The Attempt at a Solution

I think I got a right.

I took derivatives of the radius which is zero, and integrated the angular acceleration. I then just plugged the numbers into the acceleration formulas and got a_r=0 (assuming t=0) and a_θ = 5 = a.

I am a bit unsure about the 2nd one because i am having trouble getting the equation of velocity and acceleration in terms of theta. I always end up with either the angular velocity, or time variable in there. Help appreciated.

 

[Andrew Mason]

Homework Statement

A car enters the circular road with radius r = 200m at a
speed of v = 80km/h. A radar gun at O needs to rotate with constant angular
acceleration d^2θ/dt^2 = 0.025 rad/s2 to follow the motion of the car along the circular road.
a) Determine the acceleration of the car at the instant right after it enters the circular
road;
b) Determine the velocity and acceleration of the car at the instant when the radar
gun has rotated

Homework Equations

v_r = dr/dt
v_θ = r(dθ/dt)
a_r = dr/dt - r(dθ/dt)^2
a_θ = r(d^2θ/dt^2) + 2(dr/dt)θ

The Attempt at a Solution

I think I got a right.

I took derivatives of the radius which is zero, and integrated the angular acceleration. I then just plugged the numbers into the acceleration formulas and got a_r=0 (assuming t=0) and a_θ = 5 = a.

I am a bit unsure about the 2nd one because i am having trouble getting the equation of velocity and acceleration in terms of theta. I always end up with either the angular velocity, or time variable in there. Help appreciated.

I am not sure what the question is in part b). Is it the speed of the car when it has rotated 2π radians?

r is constant, since it is a circle. So you don't need to worry about dr/dt

You first have to find how long it takes the car to make the full circle: Δt. Then find the Δv in that time.

AM