A car initially traveling at 31.0 m/s undergoes a constant negative acceleration of magnitude 2.00 m/s2 after its brakes are applied.

(a) How many revolutions does each tire make before the car comes to a stop, assuming the car does not skid and the tires have radii of 0.330 m?


(b) What is the angular speed of the wheels when the car has traveled half the total distance?

vf^2=Vi^2+2ad solve for distance.

now, rotations= distance/2PIr

To find the number of revolutions each tire makes before the car comes to a stop, we need to find the total distance covered by the car.

(a) The formula to find the distance covered by an object under constant acceleration is given by:

s = (v^2 - u^2) / (2a),

where s is the distance covered, v is the final velocity, u is the initial velocity, and a is the acceleration.

First, let's find the final velocity when the car comes to a stop. The final velocity will be 0 m/s since the car comes to a stop.

Given:
Initial velocity, u = 31.0 m/s
Acceleration, a = -2.00 m/s^2
Final velocity, v = 0 m/s

Substituting the values into the formula, we have:

s = (0^2 - 31.0^2) / (2 * -2.00)

simplifying,

s = (-961 - 62) / -4.00
s = -1023 / -4.00
s = 255.75 m

The car travels a total distance of 255.75 meters before coming to a stop.

Now, let's find the number of revolutions each tire makes. The circumference of a circle is given by 2πr, where r is the radius of the circle.

Circumference of a tire = 2π * 0.330 m

The number of revolutions, n, is given by:

n = s / Circumference

Substituting the values, we have:

n = 255.75 m / (2π * 0.330 m)

n ≈ 122 revolutions

Therefore, each tire makes approximately 122 revolutions before the car comes to a stop.

(b) To find the angular speed of the wheels when the car has traveled half the total distance, we need to find the time it takes for the car to cover half the distance.

The formula to find the time of motion under constant acceleration is given by:

t = (v - u) / a,

where t is the time, v is the final velocity, u is the initial velocity, and a is the acceleration.

Given:
Initial velocity, u = 31.0 m/s
Acceleration, a = -2.00 m/s^2
Final velocity, v is unknown

Substituting the values into the formula, we have:

t = (v - 31.0) / -2.00

Let's solve for v:

v = -2.00t + 31.0

Now, half the distance is equal to s/2, which is 255.75 / 2 = 127.88 m.

Using the formula for distance:

s = ut + (1/2)at^2,

we can solve for time by substituting the known values:

127.88 = 31.0t + (1/2)(-2.00)t^2

Rearranging the equation:

-1.00t^2 + 31.0t - 127.88 = 0

Solving for t using the quadratic formula, we find:

t ≈ -6.29 s or t ≈ 20.29 s.

Since we are interested in the positive value of time, the car takes approximately 20.29 seconds to cover half the total distance.

Now, let's find the angular speed of the wheels.

Angular speed is given by the formula:

ω = v / r,

where ω is the angular speed, v is the linear velocity, and r is the radius.

Substituting the values, we have:

ω = (v / (2πr)) * 2π,

ω = (v / r).

Using the value of time, we can find the linear velocity, v, using the formula:

v = u + at,

v = 31.0 + (-2.00)(20.29),
v ≈ -9.58 m/s.

Now, substituting the values, we can find the angular speed:

ω = (-9.58 m/s) / 0.330 m ≈ -29.03 rad/s.

Therefore, the angular speed of the wheels when the car has traveled half the total distance is approximately 29.03 rad/s. Note that the negative sign represents the direction of rotation.