Jane swings on a 30m long vine initially inclined at an angle of 37o from the (downward) vertical. What is her speed at the bottom of her swing if
a) she starts from rest?
b) she pushes off with an initial speed of 4 m/s?
Help please. Show the equation and then plug in the numbers into the that equation
Thank you!
Sure! To solve this problem, we can use concepts from the conservation of mechanical energy. The total mechanical energy (E) is the sum of the kinetic energy (KE) and potential energy (PE) of the system.
a) When Jane starts from rest at the highest point of the swing (since she is on a vine initially inclined at an angle), the total mechanical energy is given by:
E = KE + PE
At the highest point, all the energy is potential energy, so:
E = PE = mgh
where m is the mass of Jane, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height above the ground.
Since Jane is on a 30 m long vine initially inclined at an angle of 37 degrees, the height above the ground can be calculated using trigonometry as:
h = 30 * sin(37)
Next, we can find the total mechanical energy using the equation:
E = mgh
Now, at the bottom of the swing, all of the initial potential energy will have been converted to kinetic energy, so the equation becomes:
E = KE
Setting the two equations equal to each other, we have:
mgh = 1/2 * mv^2
where v is the speed at the bottom of the swing.
Simplifying the equation, we get:
gh = 1/2 * v^2
Now, we can solve for the speed (v) by rearranging the equation:
v^2 = 2gh
Taking the square root of both sides, we find:
v = √(2gh)
Plugging in the known values:
g = 9.8 m/s^2
h = 30 * sin(37),
we can calculate the speed (v) at the bottom of the swing.
b) If Jane pushes off with an initial speed of 4 m/s, the equation for the total mechanical energy remains the same:
E = KE + PE
At the highest point, she has both kinetic and potential energy:
E = KE + PE = 1/2 * mv^2 + mgh
Setting the two equations equal to each other, we get:
1/2 * mv^2 + mgh = 1/2 * m(0)^2 + mgh'
where h' is the new height above the ground at the highest point.
Since Jane pushes off with an initial speed, the equation simplifies to:
1/2 * mv^2 = mgh'
Now, at the bottom of the swing where all potential energy is converted to kinetic energy, we have:
KE = 1/2 * mv^2
Setting the equations equal to each other:
1/2 * mv^2 = mgh
Rearranging the equation, we find:
v^2 = 2gh
Taking the square root of both sides, we get:
v = √(2gh)
Plugging in the known values:
g = 9.8 m/s^2
h = 30 * sin(37)
we can calculate the speed (v) at the bottom of the swing.
I hope this helps! Let me know if you have any further questions.
To find Jane's speed at the bottom of her swing, we can use the principle of conservation of mechanical energy. When Jane swings on the vine, her potential energy at the top is converted into kinetic energy at the bottom.
a) When Jane starts from rest, her initial speed is 0 m/s. Using the conservation of mechanical energy, we can equate the initial potential energy to the final kinetic energy:
Potential energy at the top = Kinetic energy at the bottom
The potential energy at the top is given by the formula:
Potential Energy = mass * gravitational acceleration * height
Since the only force acting on Jane is gravity, her mass cancels out in the equation. The gravitational acceleration is 9.8 m/s^2, and the height from the top to the bottom is the length of the vine, which is 30 m.
So, the potential energy at the top is given by:
Potential Energy = 9.8 m/s^2 * 30 m
To obtain the kinetic energy at the bottom, we use the formula:
Kinetic Energy = 1/2 * mass * velocity^2
Since we know the mass of Jane cancels out, we can write the equation as:
Potential Energy = 1/2 * velocity^2
Now, we can equate the potential and kinetic energy equations:
9.8 m/s^2 * 30 m = 1/2 * velocity^2
To find the velocity, we rearrange the equation:
velocity^2 = (9.8 m/s^2 * 30 m) / (1/2)
Simplifying the right side of the equation:
velocity^2 = 588 m^2/s^2
Finally, take the square root of both sides to find the velocity:
velocity = √(588 m^2/s^2)
b) When Jane pushes off with an initial speed of 4 m/s, we can use a similar approach. The initial kinetic energy is given by:
Initial Kinetic Energy = 1/2 * mass * initial velocity^2
Since we know the mass cancels out, we can write the equation as:
Initial Kinetic Energy = 1/2 * initial velocity^2
Again, equate the initial kinetic energy to the final kinetic energy:
Initial Kinetic Energy = 1/2 * final velocity^2
Since Jane started with an initial speed of 4 m/s, we can plug in the values:
1/2 * (4 m/s)^2 = 1/2 * (final velocity)^2
Simplifying:
8 m^2/s^2 = 1/2 * (final velocity)^2
Rearranging and solving for the final velocity:
(final velocity)^2 = 2 * 8 m^2/s^2
(final velocity)^2 = 16 m^2/s^2
Taking the square root of both sides:
final velocity = √(16 m^2/s^2)
Now you can calculate the final velocity for both cases by evaluating the equations using the given formula and plugging in the numbers.
how high was she above the bottom of the swing?
h = 30 (1-cos 37)
m g h = (1/2) m v^2 = initial Pot energy = final kinetic energy
2 * 9.81 *30 (1-cos 37) = v^2
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add (1/2)(4^4) to initial energy for part 2