One-DimensionalMotion

Physics 1

Constant Velocity

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Constant Velocity

The xd-t graph for constant velocity is linear. A common equation for any line is y = mx + b. In the graph, m is velocity (v), b is initial position (xi), and y is the final position (xf) after a time t.

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Constant Velocity

By substitution,

Subtracting xi to the left hand side,

Since xf – xi = x, this results in the expression

f ix x vt

x vt

f ix x vt

Eq. 1

Constant Acceleration

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acc

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Constant Acceleration

The v-t graph for constant acceleration is linear. A common equation for any line is y = mx + b. In the graph, m is acceleration (a), b is initial velocity (vi), and y is the final velocity (vf) after a time t. By substitution,

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atvv if Eq. 2

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Constant Acceleration

To find the displacement (x), determine the area under the v-t graph. The area can be broken into a rectangle and a triangle. The rectangle’s area is bh, where b is t and h is vi. The triangle’s are is ½bh, where b is t and h is (vf – vi).

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Constant Acceleration

The displacement is equal to the area of the rectangle and the area of the triangle.

x = area of █ + area of ▲

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2

i f i

x bh bh

x t v t v v

Constant Acceleration

Rearranging equation 2,

Substituting into the displacement equation,

Rearranging,

atvvatvv ifif

1 1

2 2 i f i ix t v t v v x t v t at

Eq. 321

2 ix v t at

Constant Acceleration

To find the displacement (x), determine the area under the v-t graph. The area is a trapezoid. The trapezoid’s area is ½(b1+b2)h, where b1 is vi and b2 is vf, and h is t.

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vi

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t Base 1

Height

Base 2

Constant Acceleration

Using the equation for the area of a trapezoid,

another equation for displacement results.

hbbarea 212

1

1

2 i fx v v t

Eq. 4

Constant Acceleration

An equation can be obtained by squaring both sides of Equation 2.

Factoring a 2a out of the last two terms,

222222 2 taatvvvatvv iifif

222

2

12 attvavv iif

Constant Acceleration

Substituting Equation 3 for the expression in parentheses,

This results in

2 2 2 2 212 2

2

f i i f iv v a v t at v v a x

2 2 2 f iv v a x Eq. 5

1-D Motion Equations

2

2

1attvx i

atvv if

2 2 2 f iv v a x

1

2 i fx v v t

vtx Eq. 1

Eq. 2

Eq. 3

Eq. 4

Eq. 5