A driver accelerates his car first at the rate of `1.8m//s^(2)` and then at the rate of `1.2m//s^(2)`. The ratio of the forces exerted by the engines will be respectively equal to

Options

A. `2:3`
B. `1:2`
C. `2:1`
D. `3:2`

Answer

To solve the problem of finding the ratio of forces exerted by the engines during two different accelerations, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between force and acceleration**:
According to Newton's second law of motion, the force exerted by an object is directly proportional to its acceleration. This can be expressed as:
\[ F = m \cdot a \]
where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration.

2. **Set up the ratio of forces**:
Since the mass of the car remains constant during both accelerations, we can express the ratio of the forces exerted by the engines during the two different accelerations as:
\[ \frac{F_1}{F_2} = \frac{m \cdot a_1}{m \cdot a_2} \]
Here, \( a_1 \) is the first acceleration (1.8 m/s²) and \( a_2 \) is the second acceleration (1.2 m/s²). The mass \( m \) cancels out, leading to:
\[ \frac{F_1}{F_2} = \frac{a_1}{a_2} \]

3. **Substitute the values of acceleration**:
Now, substitute the given values of acceleration into the equation:
\[ \frac{F_1}{F_2} = \frac{1.8 \, \text{m/s}^2}{1.2 \, \text{m/s}^2} \]

4. **Simplify the ratio**:
To simplify the ratio:
\[ \frac{F_1}{F_2} = \frac{1.8}{1.2} = \frac{18}{12} \]
Now, divide both the numerator and the denominator by 6:
\[ \frac{F_1}{F_2} = \frac{3}{2} \]

5. **State the final answer**:
Therefore, the ratio of the forces exerted by the engines is:
\[ F_1 : F_2 = 3 : 2 \]