At an instant when a particle of mass 80 g has a velocity of 25 m/s in the positive y direction, a 75g particle has a velocity of 20 m/s in the positive x direction. What is the speed of the center of mass of this two-particle system at this instant?
y momentum = .080 *25 = 2
x momentum = .075*20 = 1.5
total momentum = sqrt (4 + 2.25) = 2.5
total mass = 0.155
speed of mass center= 2.5/0.155 = 16.1
Well, center of mass can be found by taking into account the mass and velocity of each particle. So let's calculate!
First, let me grab my calculator and my comedy notebook... Ah, here they are!
Alright, let's start by finding the velocity of the center of mass in the x-direction. Since we only have one particle moving in the x-direction, we can simply use its mass and velocity. The mass of the particle is 75g, which is equivalent to 0.075 kg (remember to convert grams to kilograms!). Its velocity is 20 m/s.
Now, if you really want to make the center of mass laugh, tell it a "velocity joke". Trust me, they always find it moving! 🏃♂️
To find the velocity of the center of mass in the y-direction, we use the same approach. Since we only have one particle moving in the y-direction, we'll use its mass and velocity. The mass of the particle is 80g, which is equivalent to 0.08 kg. Its velocity is 25 m/s.
To calculate the combined velocity of the center of mass, we need to use the formula:
Vcm = (m1v1 + m2v2) / (m1 + m2)
Substituting the values:
Vcm = (0.075 kg * 20 m/s + 0.08 kg * 25 m/s) / (0.075 kg + 0.08 kg)
Now, time for a comedy break! Why did the center of mass go on a diet? Because it wanted to reduce its weight, but not its mass! 😄
Alright, back to business!
Calculating:
Vcm = (1.5 + 2) / 0.155 kg
And the answer is...
Drumroll, please!
The speed of the center of mass is approximately 25.81 m/s.
Feel free to let out a laugh and enjoy the physics of humor! 🎭😂
To find the speed of the center of mass of a two-particle system, we need to calculate the velocity of each particle relative to the center of mass, and then find the speed of the center of mass using the equation:
Speed of center of mass = square root of (Vcm_x^2 + Vcm_y^2)
Let's calculate this step-by-step:
Step 1: Calculate the velocity of each particle relative to the center of mass.
Using the principle of conservation of momentum, we can determine that the total momentum of the system is zero, since there are no external forces acting on the system.
Therefore, the momentum in the x-direction must be equal to the momentum in the y-direction:
m1 * V1_x + m2 * V2_x = 0
where m1 and m2 are the masses of the two particles, and V1_x and V2_x are their velocities in the x-direction.
Given:
m1 = 80 g = 0.08 kg (convert grams to kilograms)
V1_y = -25 m/s (negative because the velocity is in the opposite direction of the positive y-direction)
m2 = 75 g = 0.075 kg (convert grams to kilograms)
V2_x = 20 m/s (positive because the velocity is in the positive x-direction)
Plugging the values into the equation, we have:
0.08 * V1_y + 0.075 * V2_x = 0
0.08 * (-25) + 0.075 * 20 = 0
-2 + 1.5 = 0
-0.5 = 0
Therefore, we can conclude that V1_y = 25 m/s.
Step 2: Calculate the speed of the center of mass.
Now that we have the velocity of each particle relative to the center of mass, we can calculate the speed of the center of mass using the equation:
Speed of center of mass = square root of (Vcm_x^2 + Vcm_y^2)
Given:
Vcm_x = 0 (since the total momentum in the x-direction is zero)
V1_y = 25 m/s (from step 1, the velocity of particle 1 relative to the center of mass)
Plugging the values into the equation, we have:
Speed of center of mass = square root of (0^2 + 25^2)
Speed of center of mass = square root of (625)
Speed of center of mass = 25 m/s
Therefore, the speed of the center of mass of the two-particle system at this instant is 25 m/s.
To find the speed of the center of mass of a two-particle system, you need to calculate the velocity of the center of mass. The velocity of the center of mass can be calculated using the following formula:
vcm = (m1 * v1 + m2 * v2) / (m1 + m2)
Where:
- vcm is the velocity of the center of mass
- m1 and m2 are the masses of the particles
- v1 and v2 are the velocities of the particles
Given the following values:
- m1 = 80 g = 0.08 kg
- v1 = 25 m/s in the positive y direction
- m2 = 75 g = 0.075 kg
- v2 = 20 m/s in the positive x direction
First, convert the masses to kilograms:
- m1 = 0.08 kg
- m2 = 0.075 kg
Next, substitute the given values into the formula:
vcm = (0.08 kg * 25 m/s + 0.075 kg * 20 m/s) / (0.08 kg + 0.075 kg)
Simplifying the equation:
vcm = (2 kg m/s + 1.5 kg m/s) / 0.155 kg
vcm = 3.5 kg m/s / 0.155 kg
vcm = 22.58 m/s
Therefore, the speed of the center of mass of this two-particle system at this instant is approximately 22.58 m/s.