The last car unfastens from a moving train. The train continues to move at the same constant

speed. What is the ratio of the paths covered by the train and car up to the moment of the car’s
stop? Assume that the car decelerates uniformly. Solve this problem graphically.

To solve this problem graphically, we need to understand the relationship between distance, time, and velocity. Let's break down the problem into smaller steps and explain how to approach each step.

Step 1: Understand the scenario
We have a moving train with a constant speed, and at a certain point, the last car unfastens from it. The car then decelerates uniformly until it comes to a stop. We want to find the ratio of the paths covered by the train and the car up to the moment the car stops.

Step 2: Define the variables
Let's define the variables in this problem:
- d_train: Distance covered by the train from the starting point to the point the car stops.
- d_car: Distance covered by the car from the starting point to the point the car stops.
- v_train: Velocity (speed) of the train (constant).
- v_car_initial: Initial velocity (speed) of the car (same as the train's velocity).
- v_car_final: Final velocity (speed) of the car when it stops (0 m/s).

Step 3: Determine the time taken by the car to stop
Since the car decelerates uniformly until it stops, we can use the equation of motion:
v_car_final^2 = v_car_initial^2 - 2 * a * d_car
where a is the acceleration (deceleration) of the car.

In this case, v_car_initial = v_train, and v_car_final = 0.

Therefore, the equation becomes:
0 = v_train^2 - 2 * a * d_car

Since we know the final velocity is 0, we can solve for 'a':
a = v_train^2 / (2 * d_car)

Step 4: Determine the time taken by the car to stop
We can use the equation of motion again to find the time taken by the car to stop:
v_car_final = v_car_initial - a * t_car
0 = v_train - a * t_car

Solving for t_car:
t_car = v_train / a

Step 5: Determine the distance covered by the train
Since the train continues to move at a constant speed, the distance covered by the train is calculated using the formula:
d_train = v_train * t_car

Step 6: Determine the ratio of paths covered by the train and car
The ratio of the paths covered by the train and car up to the moment the car stops is given by:
Ratio = d_train / d_car

To solve this problem graphically, plot a graph with time on the x-axis and velocity on the y-axis for both the train and the car. Then calculate the areas under the curve for both the train and the car up to the moment the car stops, and find the ratio of these areas.

Remember to convert the graph into an area calculation, as the area under the curve represents distance. The ratio of the areas will give you the ratio of the paths covered by the train and the car.