You build a toy model of a water slide. A red 250 g toy car slides down a 40 degree frictionless ramp. At the bottom, just at it exists onto a horizontal part of the slide, it collides with a purple 500 g car at rest. How high should you make the water slide so that the red toy car gives the purple car a speed of 250 cm/s? Assume the collision is perfectly elastic.

Question

You build a toy model of a water slide. A red 250 g toy car slides down a 40 degree frictionless ramp. At the bottom, just at it exists onto a horizontal part of the slide, it collides with a purple 500 g car at rest. How high should you make the water slide so that the red toy car gives the purple car a speed of 250 cm/s? Assume the collision is perfectly elastic.

Answer 1

To determine the height of the water slide, we can use the principle of conservation of energy and the equation for elastic collision.

First, let's find the velocity of the red car as it exits the ramp.
Using the conservation of energy, we can equate the potential energy at the top of the ramp to the kinetic energy at the bottom:

mgh = (1/2)mv^2

Where m is the mass of the red car (250 g = 0.25 kg), g is the acceleration due to gravity (9.8 m/s^2), h is the height of the ramp, and v is the velocity of the red car.

Next, we need to find the velocity of the red car just before it collides with the purple car. This can be done using the equation for conservation of momentum:

(m_red * v_red_initial) + (m_purple * v_purple_initial) = (m_red * v_red_final) + (m_purple * v_purple_final)

Since the purple car is at rest initially (v_purple_initial = 0), the equation simplifies to:
m_red * v_red_initial = m_red * v_red_final + m_purple * v_purple_final

Substituting the given values, let's denote the velocity of the red car just before the collision as v_red_final.

(0.25 kg * v_red_initial) = (0.25 kg * v_red_final) + (0.5 kg * 250 cm/s)

Now, we have two equations. By solving them simultaneously, we can find the height of the water slide (h) and the velocity of the red car just before the collision (v_red_final).

1) mgh = (1/2)mv^2
2) (0.25 kg * v_red_initial) = (0.25 kg * v_red_final) + (0.5 kg * 250 cm/s)

By rearranging equation 2) to solve for v_red_initial:
v_red_initial = (0.25 kg * v_red_final) + (0.5 kg * 250 cm/s) / 0.25 kg

Now we substitute v_red_initial into equation 1) to find h:

(0.25 kg * g * h) = (0.5 kg * v_red_final)^2 / 2

After substituting the values of mass, acceleration due to gravity, and simplifying the equation, we can solve for h:

h = (v_red_final^2) / (2 * g)

Now we just need to substitute the desired velocity for v_red_final (250 cm/s) to find the height of the water slide:

h = (250 cm/s)^2 / (2 * 9.8 m/s^2)

After doing the calculations, we find:

h ≈ 318.88 cm

Therefore, you should make the water slide approximately 318.88 cm (or 3.19 meters) high for the red toy car to give the purple car a speed of 250 cm/s.