A freight train has a mass of 1.2 107 kg. If the locomotive can exert a constant pull of 8.4 105 N, how long does it take to increase the speed of the train from rest to 88 km/h?
To solve this problem, we need to use Newton's second law of motion, which states that the net force acting on an object equals the mass of the object multiplied by its acceleration. In this case, the net force is the pulling force exerted by the locomotive, and the acceleration is the change in speed over time.
First, we need to calculate the acceleration of the freight train. We know that the initial speed is 0 km/h because the train is at rest, and the final speed is 88 km/h. We convert these speeds to m/s (since the equation requires SI units): 0 km/h = 0 m/s and 88 km/h = 24.44 m/s.
The change in speed is the final speed minus the initial speed: 24.44 m/s - 0 m/s = 24.44 m/s.
Now, we can calculate the acceleration using Newton's second law. Rearranging the equation, we have:
acceleration = net force / mass
Given that the mass of the train is 1.2 * 10^7 kg, and the locomotive can exert a constant pull or net force of 8.4 * 10^5 N:
acceleration = (8.4 * 10^5 N) / (1.2 * 10^7 kg)
Simplifying the fractions, we get:
acceleration ≈ 0.07 m/s^2
From here, we can use the kinematic equation to find the time it takes for the train to reach its final speed:
final speed = initial speed + acceleration * time
Since the initial speed is 0 m/s, we have:
24.44 m/s = 0 m/s + (0.07 m/s^2) * time
Simplifying the equation, we get:
24.44 m/s = 0.07 m/s^2 * time
To find the time, we rearrange the equation:
time = (24.44 m/s) / (0.07 m/s^2)
Calculating this, we find:
time ≈ 348.99 s
Therefore, it takes approximately 349 seconds for the train to increase its speed from rest to 88 km/h.