How can individual force components add up to more than the magnitude of the original force vector?

Question

If I am breaking down a force $1$ N which is applied in one dimension on a mass. If the $x$-axis were collinear with the force vector, then the total force would be $1$ N along the $x$-axis and $0$ N along the $y$-axis.

When you rotate the reference frame $30$ degrees, the force can become $(1$ N$)\,\sin30^{\circ}=0.5$ N along $y$, and $(1$ N$)\,\cos30^{\circ}\approx 0.866$ N along $x$, which means the total force being applied in both directions is $1.366$ N.

Applying this same process to a velocity vector, you can also have a mass with more total kinetic energy when measured one way than when measured in the other.

Does this mean that the energy is relative to the angle that you view a object at?

Answer 1

You are incorrect in saying that, “the total force being applied in both directions is $1.366$ N.” Forces are vectors, not scalars. This means they conform to vector addition and not scalar addition.

The magnitude of the total force, $F$, is given by Pythagoras:

$$F^2 = (1\sin 30^{\circ})^2 + (1\cos 30^{\circ})^2$$

which gives $F = 1$ N, as you would expect.

Applying this same process to a velocity vector, you can also have a mass with more total kinetic energy when measured one way, than when measured in the other.

Kinetic energy is defined as the energy required to accelerate an object from rest to its current speed. Since energy is a scalar, it doesn't make to talk about the energy in a particular direction.

Does this mean that the energy is relative to the angle that you view a object at?

No. However you split up the velocity, kinetic energy is determined by the speed of the object, which does not change if you view it from different angles.