A train is traveling up a 3.6 incline at a speed of 3.30 when the last car breaks free and begins to coast without friction. How long does it take for the last car to come to rest momentarily? and how far did the last car travel before momentarily coming to rest?
To find the time it takes for the last car to come to rest, we can use the equations of motion. Let's break down the problem into two parts:
1. Part 1: The train climbing an incline.
2. Part 2: The last car coasting without friction.
For Part 1:
We know the initial speed (u = 3.30 m/s), the incline angle (θ = 3.6°), and we want to find the time taken (t). In this part, the train is moving against gravity, so we need to split the weight of the train into two components: one parallel to the incline and the other perpendicular to the incline.
The component of the weight parallel to the incline is given by W_parallel = m * g * sin(θ), where m is the mass of the train and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Let's assume the mass of the train is 1000 kg, so W_parallel = 1000 * 9.8 * sin(3.6°).
Now, we can apply Newton's second law in the horizontal direction:
W_parallel = m * a
1000 * 9.8 * sin(3.6°) = 1000 * a,
where a is the acceleration of the train.
Solving for a, we get:
a = 9.8 * sin(3.6°)
Now, we can use one of the equations of motion to find the time taken to reach the top of the incline:
v = u + a * t,
where v is the final velocity (0 m/s since it comes to rest), u is the initial velocity (3.30 m/s), a is the acceleration, and t is the time taken.
Plugging in the values, we have:
0 = 3.30 + 9.8 * sin(3.6°) * t.
Solving for t, we find the time taken to reach the top of the incline.
For Part 2:
Once the last car breaks free, it starts coasting without friction. This means that no external horizontal forces act on it, and its velocity remains constant. Therefore, the time taken for it to come to rest momentarily will be infinite.
However, to find the distance traveled by the last car before momentarily coming to rest, we can use the equation of motion:
s = u * t,
where s is the distance traveled, u is the initial velocity (3.30 m/s), and t is the time taken to reach the top of the incline.
Plugging in the values, we can calculate the distance traveled by the last car before momentarily coming to rest.
Note: The time taken to reach the top of the incline is different from the time taken to come to rest since the two parts of the problem are independent.
I hope this helps! Let me know if you have any further questions.