Megan and Catrina are at a playground swing set. Catrina pulls Megan back so that the 2.5 m long chain makes a 35 angle with the vertical. She pushes her with a speed of 6.5 m/s and Megan begins to swing forward.
a) What is Megan’s fastest speed?
b) How fast is Megan going when the chain makes a 45 with the vertical?
To find Megan's fastest speed and her speed when the chain makes a 45-degree angle with the vertical, we first need to understand the physics behind the motion of a swinging pendulum.
Let's denote Megan's initial position as point A, where she is pulled back by Catrina. At this point, the chain forms a 35-degree angle with the vertical. Megan's maximum height, point B, will occur when the chain makes a 90-degree angle with the vertical. The portion of the chain from B to A will be the radius of the circular motion.
Now, let's break down the problem step by step:
Step 1: Find the maximum height of Megan's swing (point B).
To determine Megan's maximum height, we can use the properties of a right-angled triangle. We know that the chain's length is 2.5 m, and the angle between the chain and the vertical is 35 degrees. The height of the swing at point B will be given by:
Height = Length of chain * sin(angle)
Height = 2.5 m * sin(35°)
Step 2: Find Megan's initial speed at point A.
Megan's initial speed at point A can be found using the formula:
v = sqrt(2 * g * h)
where v is the initial velocity, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height at point B.
Step 3: Find Megan's fastest speed.
To find Megan's fastest speed, we need to calculate the speed at the lowest point of the swing (point C) when the chain makes a 90-degree angle with the vertical. At this point, all of Megan's initial potential energy is converted into kinetic energy.
Step 4: Find Megan's speed at point D.
To find Megan's speed when the chain makes a 45-degree angle with the vertical, we need to find the height at point D and then use the formula:
v = sqrt(2 * g * h)
where v is the speed at point D, g is the acceleration due to gravity, and h is the height at point D.
By following these steps, we can find the answers to both questions.