A boy lining a railway bridge 49ft. high, sees a train uproaching with uniform speed and attempt to drop a stone down to railway. He releases the stone when engine is 80ft. away from the point below his position and sees the stone hits in the ground just 3 ft. in front of the engine. What is the speed of train in miles per hour?

Kinematics equation with zero initial velocity:

h=(1/2)gt^2 =>
time, t = √(2h/g) seconds
h=49', g=32.2 ft/s²
speed = distance/time
= (80-3)/t ft/s
=44.14 fps

To find the speed of the train in miles per hour, we need to use the kinematic equation for free-fall motion:

d = (1/2)gt^2,

where d is the distance traveled, g is the acceleration due to gravity (approximately 32.2 ft/s^2), and t is the time in seconds.

First, let's find the time it takes for the stone to hit the ground. The stone is dropped from a height of 49 ft, and it hits the ground just 3 ft in front of the train. So, the distance traveled by the stone is 49 - 3 = 46 ft.

Using the equation d = (1/2)gt^2, we can solve for t:

46 = (1/2)(32.2)t^2.

Multiplying through by 2 and dividing both sides by 32.2, we get:

t^2 = 2.87.

Taking the square root of both sides, we find:

t = 1.69 seconds.

Now, let's find the distance traveled by the train during this time. The train is initially 80 ft away from the point below the boy, and it covers this distance in 1.69 seconds. So, we can determine the train's speed by dividing the distance traveled by the time taken:

Speed = Distance / Time
= 80 ft / 1.69 s
≈ 47.34 ft/s.

To convert this speed to miles per hour, we can use the conversion factor that 1 mile is equal to 5280 feet and 1 hour is equal to 3600 seconds:

Speed = 47.34 ft/s * (1 mile / 5280 ft) * (3600 s / 1 hour)
≈ 32.23 miles per hour.

Therefore, the speed of the train is approximately 32.23 mph.

To find the speed of the train in miles per hour, we need to first calculate the time it takes for the stone to hit the ground, and then use that time to find the speed of the train.

Let's start by calculating the time it takes for the stone to hit the ground:

The stone is dropped from a height of 49 ft. and hits the ground 3 ft. in front of the engine. This means that the stone falls a total distance of 49 + 3 = 52 ft.

We can use the formula for the time it takes for an object to fall to calculate the time it takes for the stone to fall:

h = (1/2) * g * t^2

Where:
h = height (52 ft. in this case)
g = acceleration due to gravity (32.2 ft/s^2)
t = time

Rearranging the formula to solve for time (t):

t = sqrt((2 * h) / g)

t = sqrt((2 * 52) / 32.2)
t = sqrt(104 / 32.2)
t ≈ sqrt(3.23)
t ≈ 1.8 seconds (rounded to 1 decimal place)

Now that we have the time it takes for the stone to hit the ground, we can use it to find the speed of the train:

The stone is dropped when the engine is 80 ft. away, and it hits the ground after 1.8 seconds. Therefore, the distance covered by the train (80 ft.) is equal to the distance covered by the stone (52 ft.) plus the distance the train travels during the time the stone falls.

Let's calculate the distance traveled by the train during 1.8 seconds:

Speed = Distance / Time

Distance = Speed * Time

Distance = Speed * 1.8 seconds

80 ft = 52 ft + Speed * 1.8 seconds

Speed * 1.8 seconds = 80 ft - 52 ft

Speed * 1.8 seconds = 28 ft

Speed = 28 ft / 1.8 seconds

Now, let's convert the speed to miles per hour:

1 mile = 5280 ft
1 hour = 3600 seconds

Speed (in miles per hour) = (Speed in feet per second * 3600) / 5280

Speed (in miles per hour) = (28 ft / 1.8 seconds * 3600) / 5280

Speed (in miles per hour) ≈ 77.9 mph (rounded to 1 decimal place)

Therefore, the speed of the train is approximately 77.9 miles per hour.