Suppose a roller-coaster car starts at a height of 39 m with a speed of 2.51 m/s. What will its speed be at height = 0m if the average force of friction is equal to 0.22 times its weight? The total distance traveled is 53m.
To find the speed of the roller-coaster car at height = 0m, we can use the concept of energy conservation. The initial potential energy of the car is converted into the kinetic energy of the car at height = 0m. However, we need to take into account the work done by friction.
First, let's find the initial potential energy of the car. Since the height is given as 39m and the mass is not provided, we need to find the mass of the car using the gravitational potential energy equation.
Gravitational Potential Energy = mgh
Given that the height is 39m and acceleration due to gravity (g) is 9.8 m/s^2, we can calculate the initial potential energy:
Potential Energy = m * 9.8 * 39
Next, we need to find the work done by friction. The work done by friction is equal to the force of friction multiplied by the distance traveled. The force of friction is given as 0.22 times the weight of the car, where weight can be calculated using the formula:
Weight = mass * g
Since we have not been given the mass, we need to find it using the weight equation and mass we calculated earlier.
Now that we have the force of friction, we can calculate the work done by friction:
Work Done by Friction = Force of Friction * Distance Traveled
Finally, we equate the initial potential energy to the work done by friction plus the final kinetic energy at height = 0m to find the speed of the car at height = 0m.
Now let's calculate the answer step by step:
Step 1: Calculate the initial potential energy:
Potential Energy = m * 9.8 * 39
Step 2: Calculate the weight of the car:
Weight = m * 9.8 , where m is the mass of the car
Step 3: Calculate the force of friction:
Force of Friction = 0.22 * Weight
Step 4: Calculate the work done by friction:
Work Done by Friction = Force of Friction * Distance Traveled
Step 5: Apply the principle of conservation of energy to find the final kinetic energy:
Potential Energy = Work Done by Friction + (1/2) * m * v^2
where v is the final velocity (speed) at height = 0m.
Step 6: Solve the equation obtained in step 5 for v to find the speed of the car at height = 0m.
By following these steps, you should be able to find the speed of the roller-coaster car at height = 0m.