Question

An object is moving with a uniform acceleration a= 2t - b ms^2 determine the functions for:

A) velocity v (ms^-1) - given v (0) = 10ms^-1 ?
B) displacement s (m) - given s (0) = 5m ?
C) calculate the values of v and s for:
i) t = 2s
ii) t = 5s

b= 3/5

Answer 1

Acceleration

`a=\left(2t-(3)/(5)\right)\:\sf ms^(-2)`

Part (A)

Velocity

`\begin{aligned}\displaystyle v= \int a\:dt & =\int \left(2t-(3)/(5)\right)\:dt\\\\ & = t^2 -(3)/(5)t+C \end{aligned}`

`\begin{aligned} \textsf{As }\:v(0)=10 \implies (0)^2-(3)/(5)(0)+C & =10\\C & = 10 \end{aligned}`

`\implies v(t)=\left(t^2-(3)/(5)t+10\right)\: \sf ms^(-1)`

Part (B)

Displacement

`\begin{aligned}\displaystyle s= \int v\:dt & =\int \left(t^2-(3)/(5)t+10\right)\:dt\\\\ & = (1)/(3)t^3-(3)/(10)t^2+10t+C \end{aligned}`

`\begin{aligned} \textsf{As }\: s(0)=5 \implies (1)/(3)(0)^3-(3)/(10)(0)^2+10(0)+C & =5\\C & = 5\end{aligned}`

`\implies s(t)=\left((1)/(3)t^3-(3)/(10)t^2+10t+5\right)\: \sf m`

Part (C)

(i) t = 2 s

`v(2)=(2)^2-(3)/(5)(2)+10=(64)/(5)=12.8\: \sf ms^(-1)`

`s(2)=(1)/(3)(2)^3-(3)/(10)(2)^2+10(2)+5=(397)/(15)=26.47\: \sf m\:(2\:dp)`

(ii) t = 5 s

`v(5)=(5)^2-(3)/(5)(5)+10=32\: \sf ms^(-1)`

`s(5)=(1)/(3)(5)^3-(3)/(10)(5)^2+10(5)+5=(535)/(6)=89.17\: \sf m\:(1\:dp)`