A car is moving horizontally along a straight line with a unifrom velocity of `25 m s^-1`. A projectile is to be fired from this car in such a way that it will return to it after it has moved `100 m`. The speed of the projection must be.

Options

A. a. `10 m s^-1`
B. b.`20 m s^-1`
C. c.`15 m s^-1`
D. d. `25 m s^-1`

Explanation

To solve the problem, we need to determine the speed of the projectile that is fired from a car moving horizontally at a uniform velocity of 25 m/s, such that the projectile returns to the car after the car has moved 100 m.

### Step-by-Step Solution:

1. **Identify the Motion of the Car:**
The car is moving horizontally with a uniform velocity of 25 m/s. We need to find out how long it takes for the car to travel 100 m.

\[ \text{Time taken by the car} = \frac{\text{Distance}}{\text{Speed}} = \frac{100 \, \text{m}}{25 \, \text{m/s}} = 4 \, \text{s} \]

2. **Time of Flight of the Projectile:**
The time of flight for the projectile must also be 4 seconds because it needs to return to the car after the car has traveled 100 m.

3. **Use the Time of Flight Formula:**
The time of flight \( T \) for a projectile launched at an angle \( \theta \) with an initial speed \( u \) is given by:

\[ T = \frac{2u \sin \theta}{g} \]
where \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).

4. **Set the Time of Flight Equal to 4 Seconds:**
We know the time of flight \( T \) is 4 seconds, so we can set up the equation:

\[ 4 = \frac{2u \sin \theta}{10} \]

5. **Solve for \( u \sin \theta \):**
Rearranging the equation gives:

\[ 2u \sin \theta = 40 \]
\[ u \sin \theta = 20 \, \text{m/s} \]

6. **Determine the Speed of Projection:**
The speed of projection \( u \) can be calculated, but we need to consider that \( u \sin \theta \) gives us the vertical component of the initial velocity. Since the problem does not specify the angle of projection, we can conclude that the minimum speed of projection \( u \) must be at least \( 20 \, \text{m/s} \) when projected vertically (i.e., \( \theta = 90^\circ \)).

However, since we are looking for the speed of projection that allows the projectile to return to the car, we can consider various angles. The minimum speed of projection that satisfies the condition is \( 20 \, \text{m/s} \).

### Final Answer:
The speed of the projection must be **20 m/s**.