Derive the following kinematics equation using the position and velocity equations

[tahayassen]

Homework Statement

$${ y }_{ f }={ y }_{ i }+{ v }_{ yi }t+\frac { 1 }{ 2 } { a }_{ y }{ t }^{ 2 }\\ { v }_{ yf }={ v }_{ yi }+{ a }_{ y }t$$

Derive

$${ { v }_{ yf } }^{ 2 }={ { v }_{ yi } }^{ 2 }+2{ a }_{ y }({ y }_{ f }-{ y }_{ i })$$

2. The attempt at a solution

This isn't an actual homework question; I'm just cross-posting this thread because I'm terribly impatient and there is much more traffic here.

 

[Chestermiller]

Homework Statement

$${ y }_{ f }={ y }_{ i }+{ v }_{ yi }t+\frac { 1 }{ 2 } { a }_{ y }{ t }^{ 2 }\\ { v }_{ yf }={ v }_{ yi }+{ a }_{ y }t$$

Derive

$${ { v }_{ yf } }^{ 2 }={ { v }_{ yi } }^{ 2 }+2{ a }_{ y }({ y }_{ f }-{ y }_{ i })$$

2. The attempt at a solution

This isn't an actual homework question; I'm just cross-posting this thread because I'm terribly impatient and there is much more traffic here.

Solve for t in the second equation, and substitute it into the first equation.

 

[tahayassen]

Thanks. That derivation took much longer than expected.

 

[ehild]

You can connect the equation to work-energy theorem. The work done on a particle is equal to the change of its kinetic energy. Work done is displacement times force. Force is F=ma. $$W=(y_f-y_i)ma=1/2(mv_f^2-mv_i^2)$$. Cancel m.

But you can derive the formula easily if you remember that the displacement is average velocity multiplied by time. $$Δy=\frac{1}{2}(v_i+v_f)t$$. Substitute $t=\frac{v_f-v_i}{a}$ for t.
$$y_f-y_i=\frac{1}{2}(v_i+v_f)\frac{v_f-v_i}{a}=\frac{v_f^2-v_i^2}{2a}$$

ehild

 

[Nickie]

I would say that the derived equation is a result of applying the principles of kinematics, specifically the equations for position, velocity, and acceleration, to a one-dimensional motion in the y direction. By combining the position and velocity equations, we can eliminate the time variable and derive an equation that relates the final velocity to the initial velocity, acceleration, and displacement. This equation can be used to analyze the motion of an object in the y direction and make predictions about its final velocity based on its initial conditions. This equation is a fundamental tool in the study of motion and is applicable to a wide range of scenarios in physics and engineering.