A car moving along a circular track of radius `50.0m` acceleration from rest at `3.00 ms^(2)` Consider a situation when the car's centripetal acceleration equal its tangential acceleration

Options

A. The angle around the track does the car travel is `1` red
B. The magnitude of the car's total acceleration at that is `3sqrt(2)ms^(-1)`
C. Time elapses before this situation is `sqrt((50)/(3))s`
D. The distance travelled by the car during this time `25m`

Explanation

To solve the problem, we need to analyze the situation where a car is moving along a circular track with a radius of 50.0 m and is accelerating from rest at a rate of 3.00 m/s². We are particularly interested in the condition where the car's centripetal acceleration is equal to its tangential acceleration.

### Step-by-Step Solution:

1. **Understanding the Accelerations**:
- The tangential acceleration \( a_t \) is given as \( 3.00 \, \text{m/s}^2 \).
- The centripetal acceleration \( a_c \) is given by the formula:
\[ a_c = \frac{v^2}{r} \]
where \( v \) is the tangential velocity and \( r \) is the radius of the circular path.

2. **Setting Up the Condition**:
- We need to find the condition when \( a_c = a_t \).
- Thus, we set the two accelerations equal:
\[ \frac{v^2}{r} = a_t \]
- Substituting the known values:
\[ \frac{v^2}{50} = 3 \]

3. **Solving for Velocity \( v \)**:
- Rearranging the equation gives:
\[ v^2 = 3 \times 50 \]
\[ v^2 = 150 \]
\[ v = \sqrt{150} \approx 12.25 \, \text{m/s} \]

4. **Finding Time Elapsed**:
- The car starts from rest, so we can use the equation of motion to find the time \( t \) taken to reach this velocity:
\[ v = u + a_t t \]
- Since the initial velocity \( u = 0 \):
\[ 12.25 = 0 + 3t \]
\[ t = \frac{12.25}{3} \approx 4.08 \, \text{s} \]

5. **Finding the Distance Travelled**:
- The distance \( s \) travelled during this time can be calculated using the equation:
\[ s = ut + \frac{1}{2} a_t t^2 \]
- Again, since \( u = 0 \):
\[ s = \frac{1}{2} \times 3 \times (4.08)^2 \]
\[ s \approx \frac{1}{2} \times 3 \times 16.64 \approx 24.96 \, \text{m} \approx 25 \, \text{m} \]

6. **Finding the Angle Travelled**:
- The angle \( \theta \) in radians that the car travels can be found using the relationship:
\[ \theta = \frac{s}{r} \]
- Substituting the values:
\[ \theta = \frac{25}{50} = 0.5 \, \text{radians} \]

### Final Results:
- The distance travelled by the car is approximately **25 m**.
- The angle travelled by the car is approximately **0.5 radians**.