A small glider is placed against a compressed spring at the bottom of an air track that slopes upward at an angle of 38.0 ∘ above the horizontal. The glider has mass 6.00×10−2 kg . The spring has 630 N/m and negligible mass. When the spring is released, the glider travels a maximum distance of 1.40 m along the air track before sliding back down. Before reaching this maximum distance, the glider loses contact with the spring.

What is the kinetic energy of the glider .400m from the initial point against the compressed spring?

I used potential gravitational energy at 1.4m and at .4m, subtracting the energy to get KE, and came out with 0.36J

It was right

Well, well, well, let's calculate that kinetic energy, shall we?

To find the kinetic energy of the glider at a distance of 0.400 m from the initial point, we need to first calculate its speed at that point.

We can use the law of conservation of energy to do this. The initial potential energy of the spring is equal to the sum of the kinetic energy and potential energy at any point on the air track.

We can express the equation as:

∆PEspring = KEglider + ∆PEg

Where ∆PEspring is the change in potential energy of the spring, KEglider is the kinetic energy of the glider, and ∆PEg is the change in gravitational potential energy of the glider.

Since we're looking for the kinetic energy at a distance of 0.400 m, we can set the change in gravitational potential energy (∆PEg) equal to zero because the height difference is zero (at that point).

Now, let's put some numbers in!

The maximum distance the glider travels is 1.40 m, so the distance from the initial point that we're interested in (0.400 m) is less than that.

Using the law of conservation of energy, we have:

∆PEspring = KEglider

(1/2)kx² = (1/2)mv²

Where k is the spring constant, x is the maximum distance compressed, m is the mass of the glider, and v is the velocity of the glider.

To find v (the velocity), we can use the speed formula:

v = √(2g∆h)

Where g is the acceleration due to gravity and ∆h is the change in height.

Now, considering the angle of the air track, we can calculate the change in height (∆h):

∆h = x * sin(θ)

Where x is the distance from the initial point (0.400 m) and θ is the angle of the air track (38.0°).

Plugging in the values, we get:

∆h = 0.400 m * sin(38.0°)

Now that we know ∆h, we can calculate the velocity (v):

v = √(2 * 9.8 m/s² * ∆h)

Great! Now we have the velocity. Let's substitute it back in the earlier equation to calculate the kinetic energy (KEglider):

KEglider = (1/2)mv²

Where m is the mass of the glider (6.00 × 10⁻² kg) and v is the velocity we just found.

Plug in the numbers, and voila! You'll have the kinetic energy of the glider at a distance of 0.400 m from the initial point against the compressed spring.

Remember, laughter is the best energy booster!

To find the kinetic energy of the glider at a certain point along the air track, we need to know its velocity at that point. Given that the glider loses contact with the spring before reaching the maximum distance, we can assume that all the potential energy stored in the compressed spring is converted into kinetic energy.

To find the potential energy stored in the spring when it is compressed, we can use the formula:

Potential energy (U) = (1/2) * k * x^2

Where:
- U is the potential energy stored in the spring
- k is the spring constant (630 N/m in this case)
- x is the compression or displacement of the spring

Since we want to find the kinetic energy at a point 0.400 m from the initial point against the compressed spring, we need to find the compression of the spring at that point.

To find the compression, we can use the conservation of energy principle, which states that the initial potential energy of the system (spring) is equal to the final kinetic energy of the glider after it has traveled a certain distance (0.400 m).

So, we have:

Potential energy (initial) = Kinetic energy (final)

(1/2) * k * x^2 = (1/2) * m * v^2

Where:
- m is the mass of the glider (6.00×10−2 kg in this case)
- v is the velocity of the glider at the specified point (which we need to find)

Rearranging the equation, we can solve for v:

v = sqrt((k * x^2) / m)

Substituting the given values:
- k = 630 N/m
- x = 0.400 m
- m = 6.00×10−2 kg

v = sqrt((630 N/m * (0.400 m)^2) / (6.00×10−2 kg))

Now, we can calculate the velocity v.

After finding the velocity, we can calculate the kinetic energy at that point using the formula:

Kinetic energy = (1/2) * m * v^2