To solve this problem, we need to analyze the motion of both the train and the ball separately, and then combine their movements to find the shortest distance the ball travels before it hits the ground. Let’s break it down step by step.
Understanding the Train's Motion
The train starts from rest and accelerates at a rate of 81 km/h². First, we need to convert this acceleration into meters per second squared for easier calculations:
- 1 km/h = 1/3.6 m/s
- Therefore, 81 km/h² = 81 / 3.6 m/s² = 22.5 m/s².
Next, we can use the kinematic equation to find the final velocity of the train when it has covered a distance of 50 km (or 50,000 meters):
- Using the equation: s = ut + (1/2)at²
- Here, s is the distance (50,000 m), u is the initial velocity (0 m/s), a is the acceleration (22.5 m/s²), and t is the time.
Since the train starts from rest, we can rearrange the equation to find time:
- 50,000 = 0 + (1/2)(22.5)t²
- 50,000 = 11.25t²
- t² = 50,000 / 11.25
- t² ≈ 4444.44
- t ≈ 66.67 seconds.
Calculating the Train's Final Velocity
Now, we can find the final velocity of the train using the formula:
- v = u + at
- v = 0 + (22.5)(66.67) ≈ 1500 m/s.
Analyzing the Ball's Motion
The ball is thrown horizontally from the top of the train with a velocity of 10 m/s at a height of 4 m. To find out how far the ball travels horizontally before it hits the ground, we first need to calculate the time it takes for the ball to fall 4 meters.
Using the equation for vertical motion:
- s = ut + (1/2)gt²
- Here, s is the vertical distance (4 m), u is the initial vertical velocity (0 m/s), and g is the acceleration due to gravity (approximately 9.81 m/s²).
Rearranging gives us:
- 4 = 0 + (1/2)(9.81)t²
- 4 = 4.905t²
- t² = 4 / 4.905
- t² ≈ 0.815
- t ≈ 0.9 seconds.
Finding the Horizontal Distance
Now that we have the time it takes for the ball to fall, we can calculate how far it travels horizontally:
- Horizontal distance = horizontal velocity × time
- Horizontal distance = 10 m/s × 0.9 s = 9 m.
Calculating the Total Distance Traveled by the Ball
While the ball is falling, the train is also moving forward. In the time it takes for the ball to fall, the train travels:
- Distance = velocity × time
- Distance = 1500 m/s × 0.9 s = 1350 m.
Now, we can find the shortest distance the ball travels before hitting the ground. The ball travels horizontally 9 meters while the train moves forward 1350 meters. The total distance can be visualized as the hypotenuse of a right triangle where one leg is the horizontal distance and the other leg is the vertical distance (4 m).
Using the Pythagorean Theorem
To find the shortest distance (d) the ball travels:
- d = √(horizontal distance² + vertical distance²)
- d = √(9² + 4²) = √(81 + 16) = √97 ≈ 9.85 m.
Thus, the shortest distance the ball covers just before hitting the ground is approximately 9.85 meters. This combines both the horizontal and vertical components of its motion, giving us a complete picture of its trajectory.
