How long must a 25 degree ramp be to slow to a momentary stop a runaway truck moving along a truly horizontal highway at 85 miles per hour
To calculate the length of the ramp required to slow down the truck, we can use the principles of physics involving motion and acceleration.
Step 1: Convert the speed of the truck from miles per hour to feet per second.
To convert miles per hour to feet per second, we multiply the value by a conversion factor of 1.4667 (since 1 mile = 5280 feet and 1 hour = 3600 seconds).
85 miles per hour * 1.4667 = 124.66695 feet per second (rounded to the nearest thousandth).
Step 2: Calculate the deceleration required to stop the truck.
To determine the deceleration required, we need to know the time it takes for the truck to come to a stop. Let's assume it takes 5 seconds for the truck to stop completely.
The deceleration (a) can be calculated using the formula: a = (v_final - v_initial) / t, where v_final is the final velocity, v_initial is the initial velocity, and t is the time interval.
Since the truck needs to stop completely, the final velocity (v_final) is 0.
a = (0 - 124.66695) / 5 = -24.93339 feet per second squared.
Step 3: Calculate the distance required to decelerate.
To calculate the distance required, we can use the equation: d = (v_initial^2 - v_final^2) / (2 * a), where d is the distance, v_initial is the initial velocity, v_final is the final velocity, and a is the deceleration.
In this case, v_initial is 124.66695 feet per second, v_final is 0, and a is -24.93339 feet per second squared.
d = (124.66695^2 - 0) / (2 * -24.93339) = 316.68886 feet (rounded to the nearest thousandth).
Therefore, the ramp must be at least 316.68886 feet long to slow down the runaway truck moving along a truly horizontal highway at 85 miles per hour, assuming it takes 5 seconds to come to a stop.
To determine the length of the ramp needed to slow down a runaway truck from 85 miles per hour to a momentary stop on a 25-degree incline, we can use the principles of physics and equations of motion.
First, we need to convert the speed from miles per hour to feet per second since the units in physics equations should be consistent. We know that 1 mile is equal to 5280 feet, and 1 hour is equal to 3600 seconds. Therefore, the speed of the truck in feet per second is calculated as follows:
85 miles per hour * 5280 feet per mile / 3600 seconds per hour = 124.67 feet per second (rounded to two decimal places).
Next, we need to analyze the forces acting on the truck as it moves up the incline. The force of gravity acting on the truck will be divided into two components: one parallel to the incline and one perpendicular to it. The component parallel to the incline helps to slow down the truck, while the perpendicular component does not contribute to slowing it down.
The parallel component of the gravitational force can be calculated using the formula:
Force(parallel) = mass * gravity * sin(angle)
Where:
- mass refers to the mass of the truck
- gravity is the acceleration due to gravity (approximately 9.8 m/s² or 32.2 ft/s²)
- angle is the incline angle in radians (25 degrees converted to radians is approximately 0.436 radians)
However, before we proceed, we need to convert the speed of the truck from feet per second to meters per second since the units in the formula for force parallel will be in metric units. We know that 1 foot is approximately equal to 0.3048 meters. Therefore, the speed of the truck in meters per second is calculated as follows:
124.67 feet per second * 0.3048 meters per foot = 38.00 meters per second (rounded to two decimal places).
Now, let's calculate the parallel component of the gravitational force:
Force(parallel) = mass * gravity * sin(angle)
Force(parallel) = mass * 9.8 m/s² * sin(0.436)
To find the length of the ramp required to slow down the truck to a momentary stop, we need to equate the force parallel to the frictional force on the ramp. We can use the equation:
Force(fric) = mass * acceleration
Where:
- Force(fric) is the frictional force acting on the truck on the incline, which is equal to the force parallel
- mass refers to the mass of the truck
- acceleration is the deceleration of the truck on the incline, which we need to determine
Since we want the truck to come to a momentary stop, the acceleration of the truck is the negative value of the acceleration due to gravity (-9.8 m/s² or -32.2 ft/s²) because it opposes the direction of motion.
Force(parallel) = mass * acceleration
mass * 9.8 m/s² * sin(0.436) = mass * -9.8 m/s²
sin(0.436) = -1
However, this equation doesn't yield a valid solution. It indicates that the incline angle of 25 degrees is not steep enough to slow down the truck to a momentary stop.
If you want to find a valid solution for slowing down the truck on the incline, you would need to increase the incline angle or consider additional factors like the braking capacity of the truck, air resistance, or apply external forces to slow it down.
Ignoring friciton, the KE is has must be converted to GPE
1/2 m v^2=mgh
solve for h (convert mph to m/s first)
then, length of ramp at 25 degrees..
sin25=H/L or
length= height h above / sin25