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 solve the problem step by step, we will analyze the motion of the body moving in a circular path and determine the angle between its acceleration and velocity.

### Step 1: Identify Given Values
- Radius of the circle (R) = 80 m
- Speed of the body (V) = 20 m/s
- Rate of decrease of speed (deceleration, a_t) = 5 m/s²

### Step 2: Calculate Centripetal Acceleration
Centripetal acceleration (a_c) is given by the formula:
\[ a_c = \frac{V^2}{R} \]
Substituting the values:
\[ a_c = \frac{(20)^2}{80} = \frac{400}{80} = 5 \, \text{m/s}^2 \]

### Step 3: Identify Tangential Acceleration
The tangential acceleration (a_t) is given as:
\[ a_t = -5 \, \text{m/s}^2 \]
(Note: It is negative because the speed is decreasing.)

### Step 4: Calculate the Angle Between Acceleration and Velocity
The angle (φ) between the net acceleration and the velocity can be found using the tangent function:
\[ \tan(\phi) = \frac{a_c}{|a_t|} \]
Substituting the values:
\[ \tan(\phi) = \frac{5}{5} = 1 \]
Thus,
\[ \phi = \tan^{-1}(1) = 45^\circ \]

### Step 5: Determine the Angle Between Net Acceleration and Velocity
The angle (θ) between the net acceleration and the velocity vector is given by:
\[ \theta = 180^\circ - \phi \]
Substituting the value of φ:
\[ \theta = 180^\circ - 45^\circ = 135^\circ \]

### Final Answer
The angle made by the acceleration with its velocity is:
\[ \theta = 135^\circ \]