You build a toy model of a water slide. A red 225 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 450 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 200 cm/s? Assume the collision is perfectly elastic.

Question

You build a toy model of a water slide. A red 225 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 450 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 200 cm/s? Assume the collision is perfectly elastic.

Answer 1

(1/2) m u^2 = m g h

so u = sqrt (2 g h) (remember that, use it all the time)
so
to get u
momentum
.225 u = .225 u' + .450 kg * 2 m/s
and energy
(1/2)(.225 ) u^2 = (1/2)(.225) u'^2+ (1/2)(.450)(4)
once you have u, you can find gh

Answer 2

To determine the height of the water slide that would give the purple car a speed of 200 cm/s, we can use the principle of conservation of mechanical energy.

First, let's calculate the potential energy of the red toy car at the initial height of the water slide.

The formula for potential energy is:

Potential Energy (PE) = mass (m) * gravitational acceleration (g) * height (h)

In this case, the mass of the red toy car is 225 g, which can be converted to kilograms by dividing by 1000: 225 g / 1000 = 0.225 kg.
The gravitational acceleration (g) is approximately 9.8 m/s².
So, the potential energy (PE) at the initial height of the slide is:

PE = 0.225 kg * 9.8 m/s² * h

Next, let's calculate the kinetic energy of the red toy car at the bottom of the slide.

The formula for kinetic energy is:

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

The mass of the red toy car (m) is 0.225 kg. Since the car slides down a frictionless ramp, there is no loss of energy due to friction.
Therefore, the kinetic energy (KE) at the bottom of the slide is:

KE = (1/2) * 0.225 kg * (velocity)^2

The initial potential energy is converted into the kinetic energy at the bottom of the slide.

Since the collision is perfectly elastic, we can also use the conservation of kinetic energy to determine the final velocity of the purple car.

The initial total kinetic energy (KE) of the system (red car + purple car) is equal to the final total kinetic energy after the collision.

Initial KE = Final KE

Since the purple car is initially at rest, its initial kinetic energy is zero. Therefore:

(1/2) * 0.225 kg * (velocity)^2 = (1/2) * 0.450 kg * (200 cm/s)^2

By rearranging the equation above, we can solve for the value of (velocity)^2:

(velocity)^2 = (0.225 kg / 0.450 kg) * (200 cm/s)^2

Now, convert the velocity to meters per second (m/s) as follows:

velocity = 200 cm/s * (1 m / 100 cm) = 2 m/s

(velocity)^2 = (2 m/s)^2 = 4 m^2/s^2

Substitute this value back into the equation for potential energy and kinetic energy:

PE = 0.225 kg * 9.8 m/s^2 * h
KE = (1/2) * 0.225 kg * (2 m/s)^2

Since PE is converted into KE, we can equate the two equations:

0.225 kg * 9.8 m/s^2 * h = (1/2) * 0.225 kg * (2 m/s)^2

Simplify the equation:

9.8 m/s^2 * h = (1/2) * (2 m/s)^2

We need to solve for the height (h):

h = [(1/2) * (2 m/s)^2] / (9.8 m/s^2)

Calculate the value of h:

h = [(0.5) * 4] / 9.8

h = 0.2 m

Therefore, you should make the water slide 0.2 meters (or 20 centimeters) high so that the red toy car gives the purple car a speed of 200 cm/s.