A car of mass `1000kg` negotiates a banked curve of radius `90m` on a fictionless road. If the banking angle is `45^(@)` the speed of the car is:

Options

A. `10ms^(-1)`
B. `20ms^(-1)`
C. `30ms^(-1)`
D. `5ms^(-1)`

Explanation

To solve the problem of a car negotiating a banked curve without friction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Mass of the car, \( m = 1000 \, \text{kg} \) (not needed for calculations as it cancels out)
- Radius of the curve, \( r = 90 \, \text{m} \)
- Banking angle, \( \theta = 45^\circ \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

2. **Understand the Forces Acting on the Car:**
- The normal force \( N \) acts perpendicular to the surface of the banked road.
- The gravitational force \( mg \) acts downward.
- There is no friction, so the only forces providing the centripetal acceleration are the components of the normal force.

3. **Set Up the Equations:**
- In the vertical direction, the forces must balance:
\[ N \cos \theta = mg \]
- In the horizontal direction, the centripetal force required for circular motion is provided by the horizontal component of the normal force:
\[ N \sin \theta = \frac{mv^2}{r} \]

4. **Express Normal Force \( N \):**
- From the vertical force balance:
\[ N = \frac{mg}{\cos \theta} \]

5. **Substitute \( N \) into the Horizontal Force Equation:**
- Replace \( N \) in the horizontal equation:
\[ \frac{mg}{\cos \theta} \sin \theta = \frac{mv^2}{r} \]

6. **Simplify the Equation:**
- Cancel \( m \) from both sides:
\[ \frac{g \sin \theta}{\cos \theta} = \frac{v^2}{r} \]
- This can be rewritten using the tangent function:
\[ g \tan \theta = \frac{v^2}{r} \]

7. **Solve for \( v^2 \):**
- Rearranging gives:
\[ v^2 = g \tan \theta \cdot r \]

8. **Calculate \( \tan \theta \):**
- For \( \theta = 45^\circ \):
\[ \tan 45^\circ = 1 \]

9. **Substitute Values:**
- Now substitute \( g = 10 \, \text{m/s}^2 \), \( \tan 45^\circ = 1 \), and \( r = 90 \, \text{m} \):
\[ v^2 = 10 \cdot 1 \cdot 90 = 900 \]

10. **Find \( v \):**
- Taking the square root gives:
\[ v = \sqrt{900} = 30 \, \text{m/s} \]

### Final Answer:
The speed of the car is \( 30 \, \text{m/s} \).

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