a stone falls from a rail road overpass which is 36ft. high into the path of the train which is approaching the overpass with the uniform speed, if the stone falls when the train is 50ft. away from the overpass and the stones hits the ground just as the train arrives at the spot. how fast is the train moving?
d = Vo*t + 0.5g*t^2 = 36 Ft.
0 + 16t^2 = 36,
t^2 = 24.42,
t = 4.94 s. = Time required for stone to hit ground.
d = V*t,
V = d / t 50/4.94 = 10.1 Ft/s.
Correction: t^2 = 36/16 = 2.25,
t = 1.5 s.
V = d / t = 50/1.5 = 33.33 Ft/s.
To find the speed of the train, we need to first calculate the time it takes for the stone to fall from the overpass to the ground. We can then use this time to determine the speed of the train.
Let's assume the time it takes for the stone to fall to the ground is represented by t (in seconds).
1. Calculate the time it takes for the stone to fall:
The distance fallen by the stone is equal to the height of the overpass, which is 36ft.
Using the formula: s = ut + (1/2)gt^2, we can set u (initial velocity) and g (acceleration due to gravity) to 0, as the stone falls under the influence of gravity alone.
So, the formula simplifies to: s = (1/2)gt^2
Substituting the values, we have: 36 = 0.5 * 32.2 * t^2 (taking g as approximately 32.2 ft/s^2)
2. Solve for t:
Rearranging the equation:
72 = 32.2 * t^2
Divide both sides by 32.2:
t^2 = 2.236
Taking the square root of both sides:
t ≈ 1.496 seconds (rounded to 3 decimal places)
Now that we know the time it takes for the stone to fall to the ground, we can determine the speed of the train.
3. Calculate the speed of the train:
The stone falls when the train is 50ft away from the overpass. It takes t seconds for the stone to fall, which means that the train covers this distance in the same amount of time.
So, the speed of the train is given by: speed = distance/time
speed = 50ft/1.496s
speed ≈ 33.4 ft/s (rounded to 1 decimal place)
Therefore, the train is moving at approximately 33.4 ft/s.
To find the speed of the train, we need to use the concept of time. Let's break down the problem and solve it step by step:
1. First, we need to determine the time it takes for the stone to fall from the overpass to the ground. We can use the equation for the time of free fall: t = √(2h/g), where t is the time, h is the height, and g is the acceleration due to gravity (approximately 32 ft/s² near the Earth's surface).
Plugging in the given height of 36 ft, we have: t₁ = √(2 * 36 / 32) = 3/2 seconds.
2. Next, we need to determine the time it takes for the train to cover the distance of 50 ft from the overpass to the spot where the stone hits the ground. Let's call this time t₂.
3. Since the stone and the train arrive at the same spot at the same time, we can equate t₁ and t₂: t₁ = t₂.
4. Now, we can calculate t₂. The speed of the train is constant, so we can use the formula: speed = distance / time.
Plugging in the given distance of 50 ft and the previously calculated time of t₁, we have: t₂ = distance / speed = 50 / speed.
5. Since t₁ = t₂, we can equate the two equations: 3/2 = 50 / speed.
6. To solve for the speed, we can cross-multiply and solve the equation:
3/2 = 50 / speed
3 * speed = 2 * 50
speed = 100 / 3
speed ≈ 33.33 ft/s.
Therefore, the train is moving at a speed of approximately 33.33 ft/s.