A police van moving on a highway with a speed of 30 km h fires a bullet at a thief’s car speeding away in the same direction with a speed of 192 km h. If the muzzle speed of the bullet is 150 m s, with what speed does the bullet hit the thief’s car? (Note: Obtain that speed which is relevant for damaging the thief’s car).
The bullet hits the thief's car with a speed of 105 m s⁻¹.
Step 1: Convert speeds to same units (m s⁻¹) Speed of police van = 30 km h⁻¹ = 30 × (5/18) = 8.33 m s⁻¹ Speed of thief's car = 192 km h⁻¹ = 192 × (5/18) = 53.33 m s⁻¹ Muzzle speed of bullet = 150 m s⁻¹
Step 2: Calculate actual velocity of bullet Since the bullet is fired from a moving police van in the same direction, the actual velocity of bullet = velocity of police van + muzzle velocity Actual velocity of bullet = 8.33 + 150 = 158.33 m s⁻¹
Step 3: Calculate relative speed of impact Since both the bullet and thief's car move in the same direction, the speed with which bullet hits the car = velocity of bullet – velocity of thief's car Speed of impact = 158.33 – 53.33 = 105 m s⁻¹
This relative speed is relevant for damaging the thief's car because the damage depends on the relative velocity between the bullet and the target.
Explanation
This problem tests the concept of relative velocity in one-dimensional motion. The key insight is that the bullet inherits the velocity of the police van (since it's fired from a moving vehicle) and the damage depends on relative velocity between bullet and target. The conversion factor 5/18 is used to convert km h⁻¹ to m s⁻¹.
Solution Steps
Step 1: Convert all speeds to m s⁻¹ using factor 5/18
Step 2: Add police van velocity to muzzle velocity to get bullet's actual velocity
Step 3: Subtract thief's car velocity from bullet velocity to get relative impact speed