Kinetic Energy and Power Estimates for Linear Motion

Problem Overview

The objective of this analysis is to approximate the amount of energy and power needed for linear motion of a cart. The cart will need to travel a distance L in specified amount of time. This analysis only determines the amount of energy and power needed to accelerate and move the cart in the specified time. This is a rough analysis since it neglects friction, but it is useful for selecting a viable energy source.

Variables:

mass of car: m = 10kg (kilograms)

distance car needs to move: L = 2m (meters)

desired time of travel: t = 5s

Assumptions:

• 1. Energy loss due to friction is negligible

  2. The cart will accelerate at a constant acceleration, which requires the least amount of power to reach a designated speed.

Energy and Power Analysis

In order to determine the required energy we need to calculate the kinetic energy that will be transferred to the car, but to do this we need to find the maximum velocity of the car.

The average velocity, V_ave, is given by distance traveled over time which is:

V_ave = L/t

For constant acceleration, V=at, and the maximum final velocity is twice the average velocity:

V_max = 2 V_ave

While the car reaches distance L, it will be at V_max, and its kinetic energy is given by

E_needed = 0.5 m V_max²

The power needed is to generate the required kinetic energy in the specified time, t.

Pneeded = Eneeded/t

Force Analysis

The cart acceleration is given by the change in velocity over time, which is:

a = V_max/t

The force needed to generate this acceleration is given by:

F = ma

Typically the cart will use a motor to generate an acceleration force between the wheel and the ground. The required torque of the motor will depend on the drive train mechanism and may include factors such as wheel size and gear ratios.The specific torque can be analyzed with a Free Body Diagram. In the case of a direct attachment between a motor and a wheel the necessary torque is equal to Fr_wheel, where rwheel is the radius of the wheel. More complex drivetrains and consideration of traction between the wheel and ground will be shown in future examples.