A car 2m long and 3 m wide is moving at 10m/s when a bullet hits it in adirection making an angle of tan^-1(3/4) with the car as seen from the ground. The bullet enters one edge of the car at the corner and passes out at diagonally opposite corner. Neglecting gravity, the time for the bullet to cross the car is:

a) 1 sec

b) 0.4 sec

c) 0.2 sec

d) 0.6 sec

Ans - c

To determine the time it takes for the bullet to cross the car, we first need to analyze the situation geometrically and kinematically. The car has dimensions of 2 meters in length and 3 meters in width, and the bullet enters at one corner and exits at the diagonally opposite corner. The angle of the bullet's trajectory with respect to the car can be calculated using the tangent function.

Understanding the Bullet's Path

The angle given is tan^-1(3/4). This means that for every 4 units of horizontal distance, the bullet travels 3 units vertically. We can visualize this as a right triangle where:

• Opposite side = 3 units
• Adjacent side = 4 units
• Hypotenuse = 5 units (using Pythagoras theorem)

Calculating the Distance Across the Car

Since the bullet travels diagonally across the car, we need to find the diagonal distance from one corner of the car to the opposite corner. The diagonal distance (d) can be calculated using the Pythagorean theorem:

d = √(length² + width²)

Substituting the values:

d = √(2² + 3²) = √(4 + 9) = √13 ≈ 3.61 meters

Finding the Bullet's Velocity Component

The bullet is moving at a speed of 10 m/s. To find the effective speed of the bullet across the car, we need to determine the component of the bullet's velocity that is directed along the diagonal path. The angle of the bullet's trajectory can be calculated as:

sin(θ) = opposite/hypotenuse = 3/5

cos(θ) = adjacent/hypotenuse = 4/5

Now, the effective speed of the bullet along the diagonal can be calculated using the cosine of the angle:

V_effective = V_bullet * cos(θ) = 10 m/s * (4/5) = 8 m/s

Calculating the Time to Cross the Car

Now that we have the distance and the effective speed, we can find the time (t) it takes for the bullet to cross the car using the formula:

t = distance / speed

Substituting the values we found:

t = d / V_effective = 3.61 m / 8 m/s ≈ 0.45125 seconds

Rounding this to one decimal place gives us approximately 0.4 seconds. Therefore, the correct answer is:

0.4 seconds (Option b)

However, if we consider the options provided in the question, it seems there might be a misunderstanding in the problem statement or the options listed. Based on the calculations, the time for the bullet to cross the car is approximately 0.4 seconds, which corresponds to option b, not c.