A plane is flying horizontally 350 kph at an altitude of 420 m. At this instant, a bomb is released. How far horizontally from this point will the bomb hit the ground?
The Correct Answer and Explanation is:
To determine how far horizontally the bomb will travel before hitting the ground, we must analyze the motion using projectile motion principles. The horizontal and vertical motions of the bomb are independent of each other, except for the time they share in flight.
Step 1: Understand the Given Data
- Horizontal velocity of the plane (and the bomb): 350 km/h
- Altitude (initial vertical height): 420 meters
- Acceleration due to gravity (g): 9.8 m/s²
- Initial vertical velocity: 0 (the bomb is simply dropped, not thrown downward)
We need to determine how far horizontally the bomb travels before hitting the ground.
Step 2: Convert Units
Convert horizontal speed from km/h to m/s: 350 km/h=350×10003600=97.22 m/s350 \text{ km/h} = \frac{350 \times 1000}{3600} = 97.22 \text{ m/s}
Step 3: Determine Time to Hit the Ground
Use the vertical motion equation: h=12gt2h = \frac{1}{2} g t^2 420=12⋅9.8⋅t2420 = \frac{1}{2} \cdot 9.8 \cdot t^2 t2=420×29.8=8409.8=85.71t^2 = \frac{420 \times 2}{9.8} = \frac{840}{9.8} = 85.71 t=85.71≈9.26 secondst = \sqrt{85.71} \approx 9.26 \text{ seconds}
Step 4: Calculate Horizontal Distance
Distance=Horizontal speed×Time\text{Distance} = \text{Horizontal speed} \times \text{Time} Distance=97.22×9.26≈900 meters\text{Distance} = 97.22 \times 9.26 \approx 900 \text{ meters}
Final Answer:
The bomb will hit the ground approximately 900 meters horizontally from the release point.
Explanation
This problem is a classic example of projectile motion, where an object is launched or dropped and moves under the influence of gravity. When the bomb is released from the plane, it already has the same horizontal velocity as the plane — 350 km/h or 97.22 m/s. The bomb will continue moving horizontally at this speed (neglecting air resistance) until it hits the ground.
Meanwhile, the vertical motion is governed solely by gravity. Since the bomb starts with zero vertical velocity and accelerates downward due to gravity (9.8 m/s²), it follows a parabolic trajectory. To determine the total flight time, we calculate how long it takes for the bomb to fall from 420 meters. Using the formula for free-fall distance h=12gt2h = \frac{1}{2}gt^2, we solve for time tt and find it to be approximately 9.26 seconds.
During this time, the bomb continues moving horizontally at a constant speed. By multiplying the time in the air (9.26 s) by the constant horizontal velocity (97.22 m/s), we get the horizontal distance traveled — around 900 meters.
This result assumes there is no air resistance, which is a typical simplification in basic physics problems. In the real world, air drag would slow the bomb down slightly, so the actual distance might be a bit shorter. Nonetheless, this problem shows how the independence of vertical and horizontal motion simplifies the analysis of projectile trajectories.
