A body is moving on a circle of radius 80 m with a speed 20 m/s which is decreasing at the rate 5 `ms^(-2)` at an instant. The angle made by its acceleration with its velocity is

Options

A. `45^(@)`
B. `90^(@)`
C. `135^(@)`
D. `0^(@)`

Explanation

To find the angle made by the acceleration with the velocity of a body moving in a circular path, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Radius of the circular path (R) = 80 m
- Speed of the body (v) = 20 m/s
- Deceleration (a_t) = -5 m/s² (negative because the speed is decreasing)

2. **Calculate Centripetal Acceleration (a_c)**:
The formula for centripetal acceleration is given by:
\[ a_c = \frac{v^2}{R} \]
Substituting the values:
\[ a_c = \frac{(20 \, \text{m/s})^2}{80 \, \text{m}} = \frac{400}{80} = 5 \, \text{m/s}^2 \]

3. **Determine the Components of Acceleration**:
- The tangential acceleration (a_t) is acting in the direction opposite to the velocity because the speed is decreasing. Thus, \( a_t = -5 \, \text{m/s}^2 \).
- The centripetal acceleration (a_c) is directed towards the center of the circular path.

4. **Calculate the Angle Between Acceleration and Velocity**:
The total acceleration (a) can be represented as a vector sum of the tangential acceleration and centripetal acceleration. The angle θ between the resultant acceleration vector and the velocity vector can be found using the tangent function:
\[ \tan(\theta) = \frac{a_t}{a_c} \]
Substituting the values:
\[ \tan(\theta) = \frac{-5 \, \text{m/s}^2}{5 \, \text{m/s}^2} = -1 \]
This gives:
\[ \theta = \tan^{-1}(-1) = -45^\circ \]

5. **Determine the Angle with Respect to the Velocity**:
Since the angle θ is measured from the direction of the velocity, we need to add 90° to find the angle made by the acceleration with the velocity:
\[ \text{Angle with velocity} = 90^\circ + (-45^\circ) = 135^\circ \]

### Final Answer:
The angle made by the acceleration with its velocity is **135 degrees**.

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