A roller coaster car is moving at 20.0m/s along a straight horizontal track. what is the speed after climbing a 15.0m hill? neglect the effects of friction

The kinetic energy loss will equal the potential energy gain.

V1^2/2 - V2^2/2 = g H = 147 m^2/s^2
V1^2/2 = 200 m^2/s^2
V2^2 = 2*(200 - 147) = 106 m^2/s^2
V2 = 12.1 m/s

To determine the speed of the roller coaster car after climbing the hill, we can use the principle of conservation of mechanical energy.

The total mechanical energy of the roller coaster car is conserved, meaning that the sum of its kinetic energy and potential energy remains constant throughout the motion. At the bottom of the hill, all of the mechanical energy is in the form of kinetic energy, and at the top of the hill, all of the mechanical energy is in the form of potential energy.

Let's break down the problem and find the speed at the top of the hill:

Step 1: Calculate the initial kinetic energy (KE1) of the roller coaster car.

The kinetic energy (KE) of an object is given by the formula:
KE = (1/2) * mass * velocity^2

In this case, the roller coaster car is moving at a speed of 20.0 m/s. Since the mass is not provided in the question, we can leave it as a variable for now.

KE1 = (1/2) * mass * (20.0 m/s)^2
KE1 = 200 m^2/s^2 * mass

Step 2: Calculate the potential energy (PE2) gained by the roller coaster car after climbing the hill.

The potential energy (PE) of an object is given by the formula:
PE = mass * gravity * height

In this case, the height of the hill is given as 15.0 m, and gravity can be approximated as 9.8 m/s^2.

PE2 = mass * 9.8 m/s^2 * 15.0 m
PE2 = 147 m/s^2 * mass

Step 3: Apply the conservation of mechanical energy principle.

Since the total mechanical energy is conserved, we can equate the initial kinetic energy (KE1) to the potential energy at the top of the hill (PE2).

KE1 = PE2
200 m^2/s^2 * mass = 147 m/s^2 * mass

Step 4: Solve for the mass using the cancellation method.

Divide both sides of the equation by 147 m/s^2:
200 m^2/s^2 * mass / 147 m/s^2 = mass

Now we can cancel out the mass and solve for the remaining quantity, which is the speed of the roller coaster car at the top of the hill.

Step 5: Calculate the final speed (v2) of the roller coaster car at the top of the hill.

Plug the value of mass back into the equation for kinetic energy and solve for speed:
KE2 = (1/2) * mass * velocity^2
KE2 = (1/2) * (200 m^2/s^2 * mass) * v2^2

We know that at the top of the hill, all of the mechanical energy is in the form of potential energy, so the kinetic energy is equal to zero.

(1/2) * (200 m^2/s^2 * mass) * v2^2 = 0

Solving for v2, we can see that the speed of the roller coaster car at the top of the hill is 0 m/s.

Therefore, after climbing the 15.0 m hill, the roller coaster car comes to rest and its speed becomes 0 m/s.