A car starts from rest on a curve with a radius of 110m and accelerates at 1.2m/s^2. Through what angle will the car have traveled when the magnitude of its total acceleration is 2.1 m/s^2?
Steps please.
To find the angle through which the car will have traveled, we can use the concept of centripetal acceleration and total acceleration.
Step 1: Find the centripetal acceleration
The centripetal acceleration is given by the equation: a_c = v^2 / r, where v is the velocity and r is the radius of the curve. At any point on the curve, the centripetal acceleration is always directed towards the center of the circle.
Given that the car starts from rest, the initial velocity is 0 m/s. Therefore, the centripetal acceleration at any point on the curve can be calculated as: a_c = (v^2 - 0^2) / r = v^2 / r.
Step 2: Find the total acceleration
The total acceleration is the vector sum of the centripetal acceleration (a_c) and the tangential acceleration (a_t). The tangential acceleration is the acceleration in the direction of motion of the car.
Given that the car accelerates at 1.2 m/s^2, the tangential acceleration can be defined as: a_t = 1.2 m/s^2.
The total acceleration can be calculated as: a_total = sqrt(a_c^2 + a_t^2).
Step 3: Solve for the velocity
To solve for the velocity (v) at any point on the curve, we need to solve the equation for total acceleration:
a_total = sqrt(a_c^2 + a_t^2).
Square both sides of the equation to eliminate the square root:
a_total^2 = a_c^2 + a_t^2.
(2.1 m/s^2)^2 = (v^2 / r)^2 + (1.2 m/s^2)^2.
4.41 m^2/s^4 = (v^2 / 110 m)^2 + 1.44 m^2/s^4.
Step 4: Solve the quadratic equation
Rearrange the equation to isolate v^2:
(v^2 / 110 m)^2 = 4.41 m^2/s^4 - 1.44 m^2/s^4.
(v^2 / 110 m)^2 = 2.97 m^2/s^4.
Take the square root of both sides to solve for v:
v^2 / 110 m = sqrt(2.97 m^2/s^4).
v^2 / 110 m = 1.726 m/s^2.
Multiply both sides by 110 m to isolate v^2:
v^2 = 1.726 m/s^2 * 110 m.
v^2 = 189.86 m^2/s^2.
Take the square root of both sides to solve for v:
v = sqrt(189.86 m^2/s^2).
v ≈ 13.78 m/s.
Step 5: Calculate the angle
Now that we have the velocity, we can calculate the angle (in radians) through which the car will have traveled.
The angle can be calculated using the formula: θ = s / r, where θ is the angle in radians, s is the arc length, and r is the radius of the curve.
The arc length (s) can be calculated as: s = v * t, where t is the time taken.
Since we are not given the time, we can use the relationship between acceleration (a) and velocity (v) to find the time taken. The relationship is: a = v / t.
Rearrange the equation to solve for t:
t = v / a.
t = 13.78 m/s / 2.1 m/s^2.
t ≈ 6.56 seconds.
Now, we can calculate the arc length:
s = v * t.
s = 13.78 m/s * 6.56 seconds.
s ≈ 90.49 meters.
Finally, we can calculate the angle (θ):
θ = s / r.
θ = 90.49 m / 110 m.
θ ≈ 0.823 radians.
Therefore, the car will have traveled approximately 0.823 radians when the magnitude of its total acceleration is 2.1 m/s^2.