A 20.0 kg child on skis, initially at rest, slides 2.0 m down an incline at an angle of 20.0 degrees with the horizontal. If there is no friction between incline and skis, what is the kinetic energy of the child at the bottom of the incline?

210 j

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To determine the kinetic energy of the child at the bottom of the incline, we need to use the formula for kinetic energy:

Kinetic Energy (KE) = 1/2 * mass * velocity^2

To find the velocity of the child at the bottom of the incline, we can use the concept of conservation of energy. Initially, the child is at rest, so the potential energy is converted entirely into kinetic energy.

The potential energy (PE) of the child at the top of the incline can be calculated using the formula:

Potential Energy (PE) = mass * gravitational acceleration * height

In this case, the height is given by the vertical distance traveled along the incline, which can be calculated by multiplying the distance traveled (2.0 m) by the vertical component of the incline (sin 20°).

Now, let's calculate the potential energy:

PE = mass * gravitational acceleration * height
= 20.0 kg * 9.8 m/s^2 * (2.0 m * sin 20°)

Next, we can equate the potential energy at the top of the incline to the kinetic energy at the bottom, since there is no friction or other energy losses:

PE = KE

Now, we can substitute the formula for kinetic energy and solve for KE:

KE = 1/2 * mass * velocity^2

1/2 * mass * velocity^2 = mass * gravitational acceleration * height

Dividing both sides by mass, we get:

1/2 * velocity^2 = gravitational acceleration * height

Now, we can substitute the values we have:

1/2 * velocity^2 = 9.8 m/s^2 * (2.0 m * sin 20°)

Finally, solve for velocity by rearranging the equation:

velocity^2 = 2 * 9.8 m/s^2 * (2.0 m * sin 20°)

velocity = √(2 * 9.8 m/s^2 * (2.0 m * sin 20°))

After evaluating this expression, you can substitute the value of velocity into the formula for kinetic energy to find the answer.