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Two point objects A and B have masses 1000 kg and 3000 kg respectively. They are initially at rest with a separation equal to 1 m. Their mutual gravitational attraction then draws them together. How far from A's original position (m) will they collide___.
Correct answer is '0.75'. Can you explain this answer?
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Two point objects A and B have masses 1000 kg and 3000 kg respectively...
Since gravitational force is conservative, therefore they collide at their centre of mass —
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Two point objects A and B have masses 1000 kg and 3000 kg respectively...
Given information:
- Mass of object A = 1000 kg
- Mass of object B = 3000 kg
- Initial separation between objects A and B = 1 m
To find:
- The distance from object A's original position where the objects will collide
Assumptions:
- The objects are point masses, meaning their sizes and shapes are not considered.
- There are no external forces acting on the objects other than their mutual gravitational attraction.
Gravitational force:
The gravitational force between two objects can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
Where:
- F is the gravitational force between the objects
- G is the gravitational constant (6.673 x 10^-11 Nm^2/kg^2)
- m1 and m2 are the masses of the objects
- r is the separation between the objects
Acceleration due to gravity:
The gravitational force can also be related to the acceleration due to gravity using Newton's second law:
F = m * a
Where:
- m is the mass of the object
- a is the acceleration due to gravity
Equating the gravitational force and acceleration:
Since the objects are initially at rest, the acceleration due to gravity will cause them to move towards each other. The gravitational force acting on object A will be equal to the acceleration due to gravity multiplied by its mass:
F = m * a
G * (m1 * m2) / r^2 = m1 * a
We can rearrange this equation to solve for the acceleration:
a = G * m2 / r^2
Equation of motion:
The equation of motion for an object moving with constant acceleration is given by:
s = ut + 0.5 * a * t^2
Where:
- s is the distance traveled by the object
- u is the initial velocity of the object (which is zero in this case)
- a is the constant acceleration of the object
- t is the time taken for the object to travel the distance s
Time of collision:
To find the time taken for the objects to collide, we need to find the value of s when the collision occurs. At that point, the separation between the objects will be zero.
s = ut + 0.5 * a * t^2
0 = 0.5 * a * t^2
Since the initial velocity u is zero, the first term on the right-hand side is zero. Solving this equation gives us:
t = 0
Final separation:
Substituting the time t = 0 into the equation of motion, we can find the final separation between the objects at the time of collision:
s = ut + 0.5 * a * t^2
s = 0 * 0 + 0.5 * a * 0^2
s = 0
Therefore, the distance from object A's original position where the objects will collide is 0 meters.
- Mass of object A = 1000 kg
- Mass of object B = 3000 kg
- Initial separation between objects A and B = 1 m
To find:
- The distance from object A's original position where the objects will collide
Assumptions:
- The objects are point masses, meaning their sizes and shapes are not considered.
- There are no external forces acting on the objects other than their mutual gravitational attraction.
Gravitational force:
The gravitational force between two objects can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
Where:
- F is the gravitational force between the objects
- G is the gravitational constant (6.673 x 10^-11 Nm^2/kg^2)
- m1 and m2 are the masses of the objects
- r is the separation between the objects
Acceleration due to gravity:
The gravitational force can also be related to the acceleration due to gravity using Newton's second law:
F = m * a
Where:
- m is the mass of the object
- a is the acceleration due to gravity
Equating the gravitational force and acceleration:
Since the objects are initially at rest, the acceleration due to gravity will cause them to move towards each other. The gravitational force acting on object A will be equal to the acceleration due to gravity multiplied by its mass:
F = m * a
G * (m1 * m2) / r^2 = m1 * a
We can rearrange this equation to solve for the acceleration:
a = G * m2 / r^2
Equation of motion:
The equation of motion for an object moving with constant acceleration is given by:
s = ut + 0.5 * a * t^2
Where:
- s is the distance traveled by the object
- u is the initial velocity of the object (which is zero in this case)
- a is the constant acceleration of the object
- t is the time taken for the object to travel the distance s
Time of collision:
To find the time taken for the objects to collide, we need to find the value of s when the collision occurs. At that point, the separation between the objects will be zero.
s = ut + 0.5 * a * t^2
0 = 0.5 * a * t^2
Since the initial velocity u is zero, the first term on the right-hand side is zero. Solving this equation gives us:
t = 0
Final separation:
Substituting the time t = 0 into the equation of motion, we can find the final separation between the objects at the time of collision:
s = ut + 0.5 * a * t^2
s = 0 * 0 + 0.5 * a * 0^2
s = 0
Therefore, the distance from object A's original position where the objects will collide is 0 meters.
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Two point objects A and B have masses 1000 kg and 3000 kg respectively. They are initially at rest with a separation equal to 1 m. Their mutual gravitational attraction then draws them together. How far from A's original position (m) will they collide___.Correct answer is '0.75'. Can you explain this answer?
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