Question

Roller-coaster car speed using energy conservation. Assuming the height of the hill is 30 m, and the roller-coaster car starts from rest at the top, calculate

(a) The speed of the roller-coaster car at the bottom of the hill,

(b) At what height it will have half this speed. Take y = 0 at the bottom of the hill.

(c) What is the speed of the roller-coaster when it reaches 20 m?

Answer 1

To calculate the speed of the roller-coaster car at the bottom of the hill, we can use the principle of conservation of energy. At the top of the hill, the car has gravitational potential energy which is converted entirely into kinetic energy at the bottom of the hill. Using the formula for gravitational potential energy and kinetic energy, we can find the speed.

Step-by-step explanation:

To calculate the speed of the roller-coaster car at the bottom of the hill, we can use the principle of conservation of energy. At the top of the hill, the car has gravitational potential energy which is converted entirely into kinetic energy at the bottom of the hill. Using the formula for gravitational potential energy and kinetic energy, we can find the speed.

(a) The speed of the roller-coaster car at the bottom of the hill can be calculated using the equation:

MGH = 1/2 * mv^2

where m is the mass of the car, g is the acceleration due to gravity, h is the height of the hill, and v is the final velocity of the car. Assuming the mass of the car is negligible, g is approximately 9.8 m/s^2, and h is 30 m, we can solve for v.

(b) To find the height at which the car will have half this speed, we can use the equation:

mgh = 1/2 * m (v/2)^2

Substituting the values and solving for h will give us the desired height.

(c) Similarly, to find the speed of the roller-coaster car when it reaches 20 m, we can use the equation:

mgh = 1/2 * mv^2

Substituting the values and solving for v will give us the desired speed.

Answer 2

Using conservation of energy, the speed of the roller-coaster car at the bottom of a 30 m hill can be calculated, and the height at which it would reach half of this speed, as well as its speed at 20 m above ground.

Step-by-step explanation:

To calculate the speed of the roller-coaster car at different points of its descent using the conservation of energy principle, we need to apply the formula that equates potential energy at the top of the hill with kinetic energy at the bottom, assuming no energy is lost to friction. Given that the height of the hill is 30 m, and the gravitational constant g is approximately 9.81 m/s2, the calculations will be:

• (a) Use the equation for conservation of mechanical energy: Ep = Ek, or mgh = ½mv2. Solving for v gives us the speed at the bottom of the hill. v = √(2gh).
• (b) To find the height at half the speed, set the kinetic energy to (½)½mv2 and solve for h using the same conservation principles.
• (c) For the speed at 20 m above ground, use the height of 10 m (since y = 0 is at the bottom) in the conservation of energy formula and solve for v.