The acceleration of a bus is given by `a_x(t)=at`, where `a=1.2ms^-2`.
a. If the bus's velocity at time `t=1.0s` is `5.0ms^-1`, what is its velocity at time `t=2.0s`?
b. If the bus's position at time `t=1.0s` is `6.0m`, what is its position at time `t=2.0s`?
c. Sketch `a_x-t`, `v_x-t`, and `x-t` graphs for the motion.

Explanation

a. The velocity `v_2` at time `t`
`v_2=v_1+int_(t_1)^(t)atdt`
`=v_1+alpha/2(t^2-t_1^2)`
`=v_1-alpha/2t_1^2+alpha/2t^2`
`=(5.0ms^-1)-(0.6ms^-3)(1.0s)^2+(0.6ms^-3)t^2`
`=(4.40ms^-1)+(0.6ms^-3)t^2`.
At `t_2=2.0s`, the velocity is `v_2=(4.40ms^-1)+(0.6ms^-3)(2.0s)^2`
`=6.80ms^-1`, or `6.8ms^-1` to two significant figures.
b. the position `x_2` as a function of time is
`x_2=x_1+int_(t_1)^tv_xdt`
`=(6.0m)+int_(t_1)^t(4.40ms^-1)+(0.6ms^-3)t^(2))dt`
`=(6.0m)+(4.40ms^-1)(t-t_1)+((0.6ms^-3))/(3)(t^3-t_1^3)`
At `t=2.0s`, and with `t_1=1.0s`,
`x=(6.0m)+(4.40ms^-1)[(2.0s)-(1.0s)]`
`+(0.20ms^-3)[(2.0s)^3-(1.0s)^3]=11.8m`
c.