A cart decelerates uniformly so that its speed over a time interval (seconds) is…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch Vector spaces, linear maps and the dual: the Riesz theorem

Problem

A cart decelerates uniformly so that its speed over a time interval

[0,T]

(seconds) is modelled by

v(t)=a+bt,

with

a

in metres per second and

b

in metres per second squared. ((a)) Confirm that the set

\mathcal P_1

of such functions of

t

is a vector space under the usual addition and scalar multiplication of functions, and give the basis

\{1,t\}

. ((b)) Construct the dual basis

\{\omega^1,\omega^2\}

\mathcal P_1^*

for this basis. ((c)) The distance travelled is the linear functional

\Phi(v)=\displaystyle\int_0^T v(t)\,\mathrm dt

. Find

c_1,c_2

such that

\Phi=c_1\omega^1+c_2\omega^2

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