A 48 g ball is released from rest at h = 1.4 m above the bottom of the track shown in the figure below. It rolls down a straight 30° segment, then back up a parabolic segment whose shape is given by y = 1/4 x2, where x and y are in m. How high will the ball go on the right before reversing direction and rolling back down? Neglect friction.
Well, let's calculate that! But first, let me just say that this ball has quite the ups and downs in its life. It's like a real-life roller coaster, but without the long lines and overpriced snacks!
To find out how high the ball will go on the right side, we need to find the point where the ball reaches its highest elevation on the parabolic segment. We can do that by finding the vertex of the parabola.
The equation of the parabolic segment is given as y = (1/4)x^2. Now, we want to find the highest point, so we need to find the vertex. And what's the vertex? It's the top dog, the peak, the crème de la crème of the parabola!
The formula for the x-coordinate of the vertex is given by x = -b / 2a, where the equation is in the form of y = ax^2 + bx + c. In this case, a = 1/4 and b = 0, so the x-coordinate of the vertex is x = -0 / (2 * 1/4). Well, anything divided by zero is a big no-no in math. So, we can safely say that x = 0.
Now, let's find the corresponding y-coordinate. Plugging in x = 0 into the equation y = (1/4)x^2, we get y = (1/4)(0)^2, which simplifies to y = 0.
So, the ball reaches its highest point at x = 0 and y = 0. But wait a second, that's not very high, is it? In fact, it's not high at all! It's ground level!
Therefore, the ball doesn't go up at all on the right side. It just rolls back down, embracing its low elevation with grace and dignity.
Remember, folks, sometimes in life, you gotta go down before you can go up. And sometimes, you don't go up at all. That's the beauty of physics and the unpredictability of the universe!
To find the maximum height reached by the ball on the right side of the track, we can use the law of conservation of mechanical energy.
The initial energy of the ball at height h will be the potential energy, given by m * g * h, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height.
As the ball rolls down the track, the potential energy is converted into kinetic energy. At the bottom of the track, all the potential energy is converted into kinetic energy.
Using the law of conservation of energy, we can write:
Potential energy at height h = Kinetic energy at the bottom of the track
m * g * h = (1/2) * m * v^2
where v is the velocity of the ball at the bottom of the track.
Since the height of the track is given as h = 1.4 m, we can substitute this value into the equation:
48 g * 9.8 m/s^2 * 1.4 m = (1/2) * 48 g * v^2
Simplifying the equation, we have:
66.432 kg m^2/s^2 = 24 v^2
Dividing both sides by 24, we get:
v^2 = 2.768 kg m^2/s^2
Taking the square root of both sides, we find:
v ≈ 1.663 m/s
Now, when the ball reaches the top of the parabolic segment, its velocity will be zero. At this point, all the kinetic energy is converted into potential energy. Using the same equation as before, but substituting the velocity as zero, we can find the maximum height reached:
m * g * h_max = (1/2) * m * 0^2
48 g * 9.8 m/s^2 * h_max = 0
Dividing both sides by 48 * 9.8, we get:
h_max = 0
Therefore, the ball does not reach any height on the right side before reversing direction and rolling back down.
To determine how high the ball will go on the right before reversing direction and rolling back down, we need to analyze the energy changes of the ball.
Let's break down the problem into different segments:
1. Straight 30° Segment:
The ball rolls down the straight 30° segment, where gravitational potential energy is converted to kinetic energy. Neglecting friction, the mechanical energy is conserved. So, we can equate the initial potential energy to the final kinetic energy:
mgh = (1/2)mv^2
Here, m is the mass of the ball, g is the acceleration due to gravity, h is the initial height, and v is the velocity of the ball.
Using the given information, we can substitute the values into the equation:
(0.048 kg)(9.8 m/s^2)(1.4 m) = (1/2)(0.048 kg)v^2
Simplifying the equation, we find:
6.720 J = 0.024v^2
2. Parabolic Segment:
In the parabolic segment, the ball is subject to the force of gravity as well as the restoring force due to the shape of the curve described by the equation y = (1/4)x^2. To find the height at which the ball will stop, we need to find the x-coordinate of the highest point on the curve.
To determine the x-coordinate, we set the derivative of y with respect to x equal to zero:
dy/dx = 0
(1/4)(2x) = 0
Solving for x, we find x = 0.
This means that the highest point on the parabolic curve is at x = 0.
Next, we substitute the x-coordinate into the equation to find the y-coordinate:
y = (1/4)(0)^2
y = 0
So, the ball reaches a maximum height of 0 m on the parabolic segment.
3. Reversing Direction:
Since the ball reaches a maximum height of 0 m on the parabolic segment, it will start rolling back down immediately without gaining any additional height.
In conclusion, the ball does not go any higher on the right before reversing direction and rolling back down.