Dogs

Chapter 2. Kinematics in One Dimension

Set coordinate as x. The coordinate is used to mark positions. x: all positive and negative values, the origin is zero. Define velocity as the first derivative of x. Define acceleration as the first derivative of velocity, so the 2nd derivative of x.

���� ���� = , ���� ������ �� 2 �� ���� = = ���� ���� 2

Integrate over time, from 0 to t. At t = 0, initial position x0, initial velocity v0. Start from zero, so the use of subscript zero. At t = t, position x, velocity vx. All times, acceleration is acceleration at that instant. Not much meaning of initial acceleration. Most often no meaning at all to specify initial acceleration.

Integrate over time, from 0 to t. At t = 0, initial position x0, initial velocity v0. Start from zero, so the use of subscript zero. At t = t, position x, velocity vx.

�� 0 ��

���� ���� =

�� 0

0

���� ���� = ����

��

�� 0

���� = �� − ��0 ,

�� 0

���� ���� =

0

������ ���� = ����

������ = ���� − ��0 ,

Motion with constant acceleration, ax is constant.

�� 0

���� ���� = ���� �� = ���� − ��0 ,

�� �� ���� 0 ��

���� ���� = ��0 + ���� ��. (1)

1 ���� �� 2 , 2

��ℎ����

�� − ��0 =

= ��0 �� +

(2)

From (1), we have t = (vx – v0)/���� . Substituting into (2), we obtain

����2 − ��02 �� − ��0 = 2����

���� = ���� + ���� ∆�� ���� ���� = ��0 + ���� �� 1 ���� − ���� = ∆�� = ���� ∆�� + ���� (∆��)2 2

���� ∆�� = ���� �� +

2 ����

1 ���� �� 2 2 − ����2 = 2���� ∆��

Average velocity is a lazy-man approach. vave = (Position2 – Position1)/(Time2 – Time1). Speed is absolute value of velocity. Average speed requires too much work! Position difference is displacement. Distance can be the absolute value of displacement. Distance can also be hard to track!

Motion with constant acceleration, ax is constant.

���� = ����0 + ���� ��. (1) �� − ��0 = ����0 �� +

1 ���� �� 2 2

, (2)

From (1), we have t = (vx – v0)/���� . Substituting into (2), we obtain...