A boxcar of mass 250 tons at rest becomes uncoupled on a 4.0° grade. If the track is considered to be frictionless, what speed does the boxcar have after 11 seconds?
To determine the speed of the boxcar after 11 seconds, we need to use the principles of physics, specifically Newton's second law and the equation for acceleration on an incline.
Here's how we can solve the problem step-by-step:
1. Determine the gravitational force acting on the boxcar:
The gravitational force (Fg) can be calculated using the formula: Fg = m * g, where m is the mass and g is the acceleration due to gravity. The mass of the boxcar is given as 250 tons, which we need to convert to kilograms. 1 ton = 1000 kg, so the mass of the boxcar (m) is 250 * 1000 = 250000 kg. The acceleration due to gravity (g) is approximately 9.8 m/s^2. Therefore, Fg = 250000 kg * 9.8 m/s^2.
2. Decompose the gravitational force into two components:
Since the track is inclined at a 4.0° grade, we need to decompose the gravitational force into two components, one parallel to the incline (F_parallel) and one perpendicular to the incline (F_perpendicular). F_parallel = Fg * sin(θ), where θ is the angle of the incline.
3. Calculate the net force acting on the boxcar:
The net force (F_net) is the force acting on the boxcar parallel to the incline, which is given by: F_net = F_parallel.
4. Use Newton's second law to calculate the acceleration:
Newton's second law states that the net force acting on an object is equal to its mass times acceleration. In this case, F_net = m * a, where a is the acceleration. Rearranging the formula, we get a = F_net / m.
5. Calculate the acceleration (a) and perpendicular force (F_perpendicular):
Now that we have the net force and mass, we can calculate the acceleration. We also calculate the perpendicular force using F_perpendicular = Fg * cos(θ).
6. Calculate the distance traveled (d) on the incline after 11 seconds:
The distance traveled can be calculated using the equation: d = 0.5 * a * t^2, where t is the time (11 seconds).
7. Calculate the final speed (v) using the distance traveled:
The final speed of the boxcar can be calculated using the equation: v = sqrt(2 * a * d).
Following these steps, we can calculate the final speed of the boxcar after 11 seconds.
force down;mg*costheta
force=ma. solve for acceleration a.
then vf=a*time