Newton's 2nd Law: Displacement & Trajectory of Motion

In summary, solving a differential equation derived from Newton's 2nd law will give us a function x(t) representing displacement as a function of time. However, the graph of x(t) does not directly represent the trajectory of the motion, as it is a repeated line in one dimension and a parametric description in two or three dimensions. Only in the case of a three dimensional vector, x(t) will give the trajectory of the object. A trajectory is the path traced by the particle, and in one dimension it is a line segment while the graph x(t) represents the position of the particle.
  • #1
hisham.i
176
2
If we make a diff. equation from Newtons's 2nd law, and solve it, we will get x(t), where x(t) is the displacement as a function of time.
what do the shape of the graph x(t) means?
Does it give the trajectory of the motion?
 
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  • #2
No, in one dimesnion the trajectory will be a line repeated many times (depends on the motion).

If you go in higher dimension, f.e. 2. You will obtain two functions x(t), y(t). Both graphs again don't give you the trajectory. But if you look at (x(t),y(t)) as simply a parametric description of a curve on the xy plane, then this curve is the trajectory.
 
  • #3
If your "x(t)" is a three dimensional vector, say, [itex]\vec{x}(t)= <x(t), y(t), z(t)>, then, yes, x(t) gives the trajectory of of the object.
 
  • #4
A trajectory is the path trace out by the particle. In one dimension the path will be a line segment. A graph x against t is a representation of the position of the particle.
 

1. What is Newton's 2nd Law of Motion?

Newton's 2nd Law of Motion states that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass. This means that the greater the force applied to an object, the greater its acceleration will be, and the more massive the object, the less its acceleration will be.

2. How does displacement relate to Newton's 2nd Law?

Displacement is a measure of the change in an object's position over time. According to Newton's 2nd Law, the net force acting on an object will cause it to accelerate, thus changing its velocity and leading to a displacement.

3. What is the significance of trajectory in relation to Newton's 2nd Law?

Trajectory refers to the path that an object takes as it moves through space. In the context of Newton's 2nd Law, trajectory is important because it is affected by both the magnitude and direction of the net force acting on the object. This means that the trajectory of an object can be altered by changing the force applied to it.

4. How does friction affect the displacement and trajectory of an object?

Friction is a force that opposes motion and is present in most real-world scenarios. In terms of Newton's 2nd Law, friction can decrease the acceleration of an object, thus affecting its displacement and trajectory. For example, a car driving on a road with high friction will have a slower acceleration and a different trajectory compared to a car driving on a road with low friction.

5. Can Newton's 2nd Law be applied to objects with varying masses?

Yes, Newton's 2nd Law can be applied to objects with varying masses. This is because the law takes into account the inverse relationship between mass and acceleration. Objects with greater mass will have a lower acceleration compared to objects with smaller mass when the same amount of force is applied.

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