- #1
johncena
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In the equation v^2 = u^2 + 2aS , What kind of products are v^2 , u^2 , and aS ?
Cross product or dot product?
Cross product or dot product?
Understanding Products in the Equation v^2 = u^2 + 2aS
- Thread starter johncena
- Start date
In summary, the products v^2, u^2, and aS in the equation v^2 = u^2 + 2aS are all dot products, assuming that the force and acceleration are constant. This is because the equation can be derived using the dot product formula. However, it is important to prove this law rather than just guessing, as it may not always be the case.
- #2
Ask1122
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Please correct me if i am wrong, but i believe they will be dot products, cross products will have a change in directions as well.
- #3
Fightfish
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Definitely dot products.
When you cross a vector with itself, you get the zero vector, which is absolutely meaningless.
When you cross a vector with itself, you get the zero vector, which is absolutely meaningless.
- #4
johncena
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OK . So v^2 and u^2 are dot products ...but what about aS?
- #5
Fightfish
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Going from just a shallow point of view (without analyzing the meaning of the equation whatsoever) it must be a dot product as well as v^2 and u^2 are both scalars, which necessarily requires the product aS to yield a scalar as well.johncena said:
- #6
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Unless of course a and S are already scalars.
- #7
Gerenuk
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I think it is important to not just guess what the products might be, but rather prove the law anew. It might be none of the products. So let's do that
[tex]\Delta E_\text{kin}=\int\vec{F}\cdot\mathrm{d}\vec{s}[/tex]
[tex]\therefore m|v|^2-m|u|^2=2\vec{F}\cdot\Delta\vec{s}[/tex]
if the force is a constant
[tex]\therefore |v|^2=|u|^2+2\vec{a}\cdot\Delta\vec{s}[/tex]
or if you wish
[tex]\therefore \vec{v}\cdot\vec{v}=\vec{v}_0\cdot\vec{v}_0+2\vec{a}\cdot(\vec{s}-\vec{s}_0)[/tex]
Note that all this assumes that the force/acceleration is constant.
[tex]\Delta E_\text{kin}=\int\vec{F}\cdot\mathrm{d}\vec{s}[/tex]
[tex]\therefore m|v|^2-m|u|^2=2\vec{F}\cdot\Delta\vec{s}[/tex]
if the force is a constant
[tex]\therefore |v|^2=|u|^2+2\vec{a}\cdot\Delta\vec{s}[/tex]
or if you wish
[tex]\therefore \vec{v}\cdot\vec{v}=\vec{v}_0\cdot\vec{v}_0+2\vec{a}\cdot(\vec{s}-\vec{s}_0)[/tex]
Note that all this assumes that the force/acceleration is constant.
- #8
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[nitpick]Gerenuk said:
Just to clarify, Gerenuk means that force and acceleration are both constant.
[tex]\frac{force}{acceleration}[/itex] is the same as the mass, which is always constant (at nonrelativistic speeds)
[/nitpick]
FAQ: Understanding Products in the Equation v^2 = u^2 + 2aS
What is the difference between dot product and cross product?
The dot product is a mathematical operation that takes two vectors and produces a scalar value. It is calculated by multiplying the corresponding components of the two vectors and then adding the products together. The cross product, on the other hand, is a vector operation that takes two vectors and produces a third vector that is perpendicular to both of the original vectors. It is calculated by taking the cross product of the two vectors and multiplying them by the sine of the angle between them.
What are the geometric interpretations of dot product and cross product?
The dot product has a geometric interpretation as the projection of one vector onto another vector. This means that it is a measure of how much of one vector lies in the direction of the other vector. The cross product has a geometric interpretation as the area of the parallelogram formed by the two vectors. This means that it is a measure of the magnitude of the two vectors and the angle between them.
Are dot product and cross product commutative?
No, both dot product and cross product are not commutative. This means that the order of the vectors matters when performing these operations. The dot product is commutative only when the vectors are parallel to each other, while the cross product is never commutative.
In what situations are dot product and cross product used?
The dot product is commonly used in physics and engineering to calculate work, force, and energy. It is also used in computer graphics for lighting and shading calculations. The cross product is used in physics, engineering, and computer graphics to calculate torque, angular momentum, and surface normal vectors.
How are dot product and cross product related to vector multiplication?
The dot product is a type of vector multiplication called the scalar product, while the cross product is a type of vector multiplication called the vector product. Both operations are used to combine two vectors and produce a new vector or scalar.