A car starts from rest on a curve with a radius of 130 m and accelerates at 0.600 m/s^2. How many revolutions will the car have gone through when the magnitude of its total acceleration is 2.90 m/s^2?
To find the number of revolutions the car will have gone through, we need to determine the speed of the car at the point when its total acceleration is 2.90 m/s^2.
Given:
Radius of the curve (r) = 130 m
Acceleration (a) = 0.600 m/s^2
Magnitude of total acceleration (|a|) = 2.90 m/s^2
The total acceleration of the car at any point on the curve is the vector sum of two components:
1. Centripetal acceleration (ac): directed towards the center of the curve
2. Tangential acceleration (at): directed along the tangent to the curve
Equation for centripetal acceleration:
ac = v^2 / r
Equation for tangential acceleration:
at = a
To find the speed of the car at the point where |a| = 2.90 m/s^2, we can use the Pythagorean theorem:
|a| = √(ac^2 + at^2)
Squaring both sides of the equation to eliminate the square root:
a^2 = ac^2 + at^2
Now we can solve for the tangential acceleration:
at^2 = a^2 - ac^2
Substituting the given values:
at^2 = (2.90 m/s^2)^2 - (0.600 m/s^2)^2
at^2 = 8.41 m^2/s^4 - 0.360 m^2/s^4
at^2 = 8.05 m^2/s^4
Taking the square root of both sides:
at = √(8.05 m^2/s^4)
at ≈ 2.841 m/s^2
Now that we have the tangential acceleration, we can find the speed of the car by integrating the tangential acceleration with respect to time:
at = dv/dt
Integrating both sides:
∫ at dt = ∫ dv
∫ 2.841 dt = ∫ dv
2.841t = v
To find the time it takes for the car to reach this speed, we can use the equation:
v = at
2.841t = 2.841 m/s^2 * t
t cancels out:
1 = 2.841 m/s^2
The tangential distance traveled (ds) can be calculated using the equation:
ds = v dt
ds = 2.841 m/s * dt
ds = 2.841 dt (since the car starts from rest)
To find the number of revolutions (n) the car has gone through, we need to relate the tangential distance to the circumference of the circle:
ds = n * circumference
n * circumference = 2.841 dt
n = (2.841 dt) / circumference
The circumference of a circle is given by:
circumference = 2 * π * r
Substituting the value of r:
circumference = 2 * π * 130 m
circumference ≈ 817.8 m
Substituting this value, the equation becomes:
n ≈ (2.841 dt) / 817.8 m
Now we need to find the time (dt) it took for the car to reach this speed. We know that the car starts from rest, so its initial speed (vi) is 0 m/s. The final speed (vf) can be calculated using the equation:
vf = vi + at
vf = 0 m/s + 2.841 m/s^2 * t
vf = 2.841 m/s^2 * t
Solving for t:
t = vf / 2.841 m/s^2
t = 2.841t / 2.841 m/s^2
t = 1 s
Now, substituting the value of dt and the calculated time (t):
n ≈ (2.841 s) / 817.8 m
n ≈ 0.00347 revolutions
Therefore, the car will have gone through approximately 0.00347 revolutions when the magnitude of its total acceleration is 2.90 m/s^2.
To solve this problem, we need to find the time it takes for the car to reach an acceleration of 2.90 m/s^2, and then use that time to calculate the number of revolutions.
Step 1: Determine the time it takes to reach an acceleration of 2.90 m/s^2.
To find the time, we can use the formula for acceleration:
a = Δv / t
Rearranging for time:
t = Δv / a
Given:
Initial acceleration (a1) = 0.600 m/s^2
Final acceleration (a2) = 2.90 m/s^2
We can now calculate the time it takes to accelerate from 0.600 m/s^2 to 2.90 m/s^2:
t = (2.90 - 0.600) m/s^2 / 0.600 m/s^2
t = 2.30 m/s^2 / 0.600 m/s^2
t ≈ 3.83 seconds
Step 2: Calculate the number of revolutions.
We can use the formula for centripetal acceleration to determine the speed of the car once it reaches an acceleration of 2.90 m/s^2:
a_c = v^2 / r
where
a_c = centripetal acceleration
v = linear speed of the car
r = radius of the curve
Rearranging for linear speed:
v = √(a_c * r)
Given:
Centripetal acceleration (a_c) = 2.90 m/s^2
Radius (r) = 130 m
Now, we can calculate the linear speed of the car using the given centripetal acceleration and radius:
v = √(2.90 m/s^2 * 130 m)
v = √(377 m^2/s^2)
v ≈ 19.4 m/s
Next, we need to calculate the distance the car travels during the acceleration phase:
s = v_1 * t + (1/2) * a * t^2
where
s = distance
v_1 = initial velocity
t = time
a = acceleration
Given:
Initial velocity (v_1) = 0 m/s
Acceleration (a) = 0.600 m/s^2
Time (t) = 3.83 seconds
Substituting the values into the equation:
s = 0 m/s * 3.83 s + (1/2) * 0.600 m/s^2 * (3.83 s)^2
s = 0 + 0.290 m * (14.7 s^2)
s ≈ 4.25 m
Finally, we can calculate the number of revolutions using the formula:
Number of revolutions = s / (2πr)
Substituting the values:
Number of revolutions = 4.25 m / (2π * 130 m)
Number of revolutions ≈ 0.0099
Therefore, the car will have gone through approximately 0.0099 revolutions when the magnitude of its total acceleration reaches 2.90 m/s^2.