Linear Kinematics

Chapter 3

Definition of Kinematics

• Kinematics is the description of motion. Motion is described using position, velocity and acceleration.

• Position, velocity and acceleration are all vector quantities.

Velocity• Velocity is defined as the rate of change in position,

or the slope of the position – time graph. The units for velocity are m/s.

𝑉=𝑃 𝑓 − 𝑃𝑖

Δ𝑡

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦=∆ 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛∆𝑇𝑖𝑚𝑒

Relationship between Slope and Velocity

• If the slope is horizontal the velocity must be zero.

• If the slope is upward the velocity must be positive.

• If the slope is downward the velocity must be negative.

• Notice that point 1 has less slope than point 2 & 3, compare there velocities.

Taking a Derivative

• The process of evaluating the slope to get the rate of change is called taking a derivative.

• The rules for estimating velocity from position are:

1.If the slope is horizontal, the velocity is 0.2.If the slope is positive (up), the velocity is

positive.3.If the slope is negative (down), the velocity is

negative.

Acceleration• Acceleration is defined as the rate of change in

velocity, or the slope of the velocity – time graph.• The units for acceleration are m/s2.

𝐴=𝑉 𝑓 −𝑉 𝑖

Δ𝑡

𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛=∆𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦∆𝑇𝑖𝑚𝑒

• If the slope of velocity is horizontal the acceleration must be zero.

• If the slope of velocity is upward the acceleration must be positive.

• If the slope of velocity is downward the acceleration must be negative.

Acceleration is the Derivative of Velocity

• The rules for estimating acceleration from velocity are:

1. If the slope of velocity is horizontal, the acceleration is 0.

2. If the slope of velocity is positive (up), the acceleration is positive.

3. If the slope of velocity is negative (down), the acceleration is negative.

Integration• Integration is the mathematical process of getting the

area underneath a curve.• Integration of acceleration gives the change in

velocity.• Integration of velocity gives the change in position.• The integral sign can be interpreted as get the area

underneath the curve.

𝑉∫𝑡0

𝑡1

𝐴𝑑𝑡

∫The change in velocity over the interval from t0 to t1 is equal to the area underneath the acceleration – time curve.

Integration of AccelerationThere are several methods of integration. Determining the area of a rectangle is one method of integration. Area = (Height × Width) + Initial Value

V = Height x Width

V = (2 m/s2)(2 s)

V = 4 m/s

Over the interval from t = 0 to t = 2 s the velocity must change by +4 m/s.

Area = 4 m/s, velocity changes by 4 m/s.

Integration of Acceleration

V = Height x Width

V = (−3 m/s2)(3 s)

V = −9 m/s

Over the interval from t = 2 to t = 5 s the velocity must change by −9 m/s.

Area = 9 m/s, velocity changes by 9 m/s.

Integration of Acceleration

V = Height x Width

V = (4 m/s2)(2 s)

V = 8 m/s

Over the interval from t = 5 to t = 7 s the velocity must change by +8 m/s.

Area = 8 m/s, velocity changes by 8 m/s.

Integration of VelocityThe integration of velocity gives the change in position.

P = Height x Width

P = (−3 m/s)(2 s)

P = −6 m

Over the interval from t = 0 to t = 2 s the position must change by −6 m.

Area = −6 m, position changes by −6 m.

Integration of VelocityThe integration of velocity gives the change in position.

P = Height x Width

P = (2 m/s)(3 s)

P = +6 m

Over the interval from t = 2 to t = 5 s the position must change by +6 m.

Area = +6 m, position changes by +6 m.

Integration of VelocityThe integration of velocity gives the change in position.

P = Height x Width

P = (−4 m/s)(3 s)

P = −12 m

Over the interval from t = 5 to t = 8 s the position must change by −12 m.

Area = −12 m, position changes by −12 m.

Evaluate slope to estimate velocity Evaluate area to estimate position

A zero for velocity is a local max or min in position

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦=∆ 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛∆𝑇𝑖𝑚𝑒 𝑉=

𝑃 𝑓 −𝑃𝑖

Δ𝑡=(B3 − B2)/0.1

Computing Velocity from Position in Excel

Excel Filename: Get Vel & Accel Data Set 1.xls

𝐴=𝑉 𝑓 −𝑉 𝑖

Δ𝑡=(C4 − C3)/0.1

Computing Acceleration from Velocity in Excel

Excel Filename: Get Vel & Accel Data Set 1.xls

𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛=∆𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦∆𝑇𝑖𝑚𝑒

Integration of Acceleration in Excel

∆𝑉=∫ 𝐴𝑑𝑡𝑉 𝑓 −𝑉 𝑖=∫ 𝐴×𝑑𝑡

𝑉 𝑓=∫(𝐴×𝑑𝑡)+¿𝑉 𝑖¿Velocity Final = (Acceleration Time) + Velocity Initial

The general equation for integration is:

Area = (Height Width) + Initial Value

Integration of Acceleration in Excel

𝑉 𝑓=∫(𝐴×𝑑𝑡)+¿𝑉 𝑖¿Area = (Height Width) + Initial Value

=(D4 * 0.1) + E3

Integration of Velocity in Excel

∆ 𝑃=∫𝑉 𝑑𝑡𝑃 𝑓 − 𝑃 𝑖=∫𝑉 ×𝑑𝑡

𝑃 𝑓=∫(𝑉 ×𝑑𝑡 )+¿𝑃 𝑖¿Position Final = (Velocity Time) + Position Initial

The general equation for integration is:

Area = (Height Width) + Initial Value

Integration of Velocity in Excel

𝑃 𝑓=∫(𝑉 ×𝑑𝑡 )+¿𝑃 𝑖¿Area = (Height Width) + Initial Value

=(E3 * 0.1) + F2

Relationship between Acceleration & Velocity

Relationship between Velocity & Position

What Does The Initial Value Do?

𝑃 𝑓=∫(𝑉 ×𝑑𝑡 )+¿𝑃 𝑖¿ Area = (Height Width) + Initial Value

The initial value tells you where to start. It simply moves the curve up or down on the Y axis.