A freight train has a mass of 1.0 10^7 kg. If the locomotive can exert a constant pull of 7.5 10^5 N, how long does it take to increase the speed of the train from rest to 66 km/h?
44.44sec
To find the time it takes to increase the speed of the train, we can use Newton's second law of motion and the equation of motion.
Step 1: Convert the speed from km/h to m/s.
Given speed = 66 km/h
1 km = 1000 m
1 hour = 3600 s
66 km/h = (66 × 1000) / 3600 m/s
Speed = 18.33 m/s (approximately)
Step 2: Apply Newton's second law of motion.
The net force acting on the train is equal to the mass of the train times its acceleration.
Net force = mass × acceleration
Step 3: Calculate the acceleration of the train.
We need to find the acceleration of the train by rearranging the equation.
acceleration = (net force) / (mass)
Given net force (F) = 7.5 × 10^5 N
Given mass (m) = 1.0 × 10^7 kg
acceleration = (7.5 × 10^5 N) / (1.0 × 10^7 kg)
acceleration = 0.075 m/s² (approximately)
Step 4: Use the equation of motion to find the time.
The equation of motion is:
final velocity = initial velocity + (acceleration × time)
Given initial velocity (u) = 0 m/s (train at rest)
Given final velocity (v) = 18.33 m/s
Given acceleration (a) = 0.075 m/s²
Rearrange the equation to solve for time (t):
t = (v - u) / a
t = (18.33 m/s - 0 m/s) / 0.075 m/s²
t = 244.4 seconds (approximately)
Therefore, it takes approximately 244.4 seconds for the train to increase its speed from rest to 66 km/h.
To determine the time it takes to increase the speed of the train, we can use Newton's second law of motion, which states that the force applied to an object is equal to the mass of the object multiplied by its acceleration (F = m * a).
In this case, the force applied by the locomotive is known to be 7.5 * 10^5 Newtons (N), and the mass of the train is given as 1.0 * 10^7 kilograms (kg). We need to find the acceleration to calculate the time it takes for the train to reach the desired speed.
To find the acceleration, we can rearrange Newton's second law equation as follows:
a = F / m
Substituting the given values into the equation, we have:
a = (7.5 * 10^5 N) / (1.0 * 10^7 kg)
Now, let's calculate the acceleration:
a = 7.5 * 10^5 / 1.0 * 10^7 = 0.075 m/s^2
Next, we need to convert the final speed from km/h to m/s:
66 km/h = (66 * 1000) m / 3600 s = 18.33 m/s
Now we can use the kinematic equation to find the time it takes for the train to reach this speed:
v = u + at
Given that the initial velocity (u) is zero because the train starts from rest, and the final velocity (v) is 18.33 m/s, and the acceleration (a) is 0.075 m/s^2, the equation becomes:
18.33 = 0 + (0.075)t
Simplifying the equation:
18.33 = 0.075t
Now, divide both sides of the equation by 0.075 to solve for t:
t = 18.33 / 0.075 ≈ 244.4 seconds
Therefore, it takes approximately 244.4 seconds for the train to increase its speed from rest to 66 km/h.