To solve this problem, we need to use the principles of energy conservation. The total energy of the system (the car and the spring) is conserved, meaning the initial energy of the car is equal to the final energy of the spring.

Step 1: Calculate the initial potential energy of the car at the top of the incline. This is given by the formula PE = mgh, where m is the mass of the car, g is the acceleration due to gravity, and h is the height of the incline.

The height can be calculated using the sine of the incline angle (10 degrees) and the length of the incline (50 m). So, h = 50sin(10) = 8.7 m.

So, PE = 2000 kg * 9.8 m/s^2 * 8.7 m = 170,360 Joules.

Step 2: At the bottom of the incline, all of this potential energy will have been converted into kinetic energy, which is given by the formula KE = 1/2 mv^2. We can set this equal to the potential energy to find the velocity of the car at the bottom of the incline:

170,360 J = 1/2 * 2000 kg * v^2

Solving for v gives v = sqrt((2*170360 J) / 2000 kg) = 18.5 m/s.

Step 3: When the car hits the spring, all of its kinetic energy will be converted into potential energy stored in the spring, which is given by the formula PE = 1/2 kx^2, where k is the spring constant and x is the distance the spring is compressed (2 m in this case).

Setting this equal to the kinetic energy and solving for k gives:

1/2 * 2000 kg * (18.5 m/s)^2 = 1/2 * k * (2 m)^2

Solving for k gives k = (2000 kg * (18.5 m/s)^2) / (2 m)^2 = 85.1 kN/m.

So, the answer is (b) 85.1 kN/m.

Question

To solve this problem, we need to use the principles of energy conservation. The total energy of the system (the car and the spring) is conserved, meaning the initial energy of the car is equal to the final energy of the spring.

Step 1: Calculate the initial potential energy of the car at the top of the incline. This is given by the formula PE = mgh, where m is the mass of the car, g is the acceleration due to gravity, and h is the height of the incline.

The height can be calculated using the sine of the incline angle (10 degrees) and the length of the incline (50 m). So, h = 50sin(10) = 8.7 m.

So, PE = 2000 kg * 9.8 m/s^2 * 8.7 m = 170,360 Joules.

Step 2: At the bottom of the incline, all of this potential energy will have been converted into kinetic energy, which is given by the formula KE = 1/2 mv^2. We can set this equal to the potential energy to find the velocity of the car at the bottom of the incline:

170,360 J = 1/2 * 2000 kg * v^2

Solving for v gives v = sqrt((2*170360 J) / 2000 kg) = 18.5 m/s.

Step 3: When the car hits the spring, all of its kinetic energy will be converted into potential energy stored in the spring, which is given by the formula PE = 1/2 kx^2, where k is the spring constant and x is the distance the spring is compressed (2 m in this case).

Setting this equal to the kinetic energy and solving for k gives:

1/2 * 2000 kg * (18.5 m/s)^2 = 1/2 * k * (2 m)^2

Solving for k gives k = (2000 kg * (18.5 m/s)^2) / (2 m)^2 = 85.1 kN/m.

So, the answer is (b) 85.1 kN/m.

Knowee AI · Accepted Answer