derive the formula to calculate kinetic energy
Answers
Answered by
1
hii.... here is ur ans
Attachments:
Answered by
0
How to Derive the Formula for Kinetic Energy
When there are no opposing forces, a moving body tends to keep moving with a steady velocity as we know from Newton's first law of motion. If, however, a resultant force does act on a moving body in the direction of its motion, then it will accelerate per Newton's second law {\mathbf {F}}=m{\mathbf {a}}. The work done by the force will become converted into increased kinetic energy in the body. We derive the expression for kinetic energy from these basic principles.
Method One of Two:
Derivation Using Calculus
1
Begin with the Work-Energy Theorem.The work that is done on an object is related to the change in its kinetic energy.\Delta K=W
2
Rewrite work as an integral. The end goal is to rewrite the integral in terms of a velocity differential.\Delta K=\int {\mathbf {F}}\cdot {\mathrm {d}}{\mathbf {r}}
3
Rewrite force in terms of velocity. Note that mass is a scalar and can therefore be factored out.{\begin{aligned}\Delta K&=\int m{\mathbf {a}}\cdot {\mathrm {d}}{\mathbf {r}}\\&=m\int {\frac {{\mathrm {d}}{\mathbf {v}}}{{\mathrm {d}}t}}\cdot {\mathrm {d}}{\mathbf {r}}\end{aligned}}
4
Rewrite the integral in terms of a velocity differential. Here, it is trivial, because dot products commute. Recall the definition of velocity as well.{\begin{aligned}\Delta K&=m\int {\frac {{\mathrm {d}}{\mathbf {r}}}{{\mathrm {d}}t}}\cdot {\mathrm {d}}{\mathbf {v}}\\&=m\int {\mathbf {v}}\cdot {\mathrm {d}}{\mathbf {v}}\end{aligned}}
5
Integrate over change in velocity.Typically, initial velocity v_{{0}} is set to 0.{\begin{aligned}\Delta K&={\frac {1}{2}}mv^{{2}}-{\frac {1}{2}}mv_{{0}}^{{2}}\\&={\frac {1}{2}}mv^{{2}}\end{aligned}}
When there are no opposing forces, a moving body tends to keep moving with a steady velocity as we know from Newton's first law of motion. If, however, a resultant force does act on a moving body in the direction of its motion, then it will accelerate per Newton's second law {\mathbf {F}}=m{\mathbf {a}}. The work done by the force will become converted into increased kinetic energy in the body. We derive the expression for kinetic energy from these basic principles.
Method One of Two:
Derivation Using Calculus
1
Begin with the Work-Energy Theorem.The work that is done on an object is related to the change in its kinetic energy.\Delta K=W
2
Rewrite work as an integral. The end goal is to rewrite the integral in terms of a velocity differential.\Delta K=\int {\mathbf {F}}\cdot {\mathrm {d}}{\mathbf {r}}
3
Rewrite force in terms of velocity. Note that mass is a scalar and can therefore be factored out.{\begin{aligned}\Delta K&=\int m{\mathbf {a}}\cdot {\mathrm {d}}{\mathbf {r}}\\&=m\int {\frac {{\mathrm {d}}{\mathbf {v}}}{{\mathrm {d}}t}}\cdot {\mathrm {d}}{\mathbf {r}}\end{aligned}}
4
Rewrite the integral in terms of a velocity differential. Here, it is trivial, because dot products commute. Recall the definition of velocity as well.{\begin{aligned}\Delta K&=m\int {\frac {{\mathrm {d}}{\mathbf {r}}}{{\mathrm {d}}t}}\cdot {\mathrm {d}}{\mathbf {v}}\\&=m\int {\mathbf {v}}\cdot {\mathrm {d}}{\mathbf {v}}\end{aligned}}
5
Integrate over change in velocity.Typically, initial velocity v_{{0}} is set to 0.{\begin{aligned}\Delta K&={\frac {1}{2}}mv^{{2}}-{\frac {1}{2}}mv_{{0}}^{{2}}\\&={\frac {1}{2}}mv^{{2}}\end{aligned}}
Similar questions