Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 9: END-OF-CHAPTER REVIEW EXERCISE 6

A vehicle is moving in a straight line. The velocity $v \mathrm{m}{ \mathrm{s}}^{-1}$ at time $t \mathrm{s}$ after the vehicle starts is given by $v=A\left(t-0.05t^2\right)$ for $0⩽t⩽15$ and $v=\frac{B}{t^2}$ for $t⩾15$, where $A$ and $B$ are constants. The distance travelled by the vehicle between $t=0 \mathrm{s}$ and $t=15 \mathrm{s}$ is $225 \mathrm{m}$.

Find the value of $A$ and show that $B=3375$.

A vehicle is moving in a straight line. The velocity $v \mathrm{m}{ \mathrm{s}}^{-1}$ at time $t \mathrm{s}$ after the vehicle starts is given by $v=A\left(t-0.05t^2\right)$ for $0 \mathrm{s}⩽t⩽15 \mathrm{s}$ and $v=\frac{B}{t^2}$ for $t⩾15$, where $A$ and $B$ are constants. The distance travelled by the vehicle between $t=0 \mathrm{s}$ and $t=15 \mathrm{s}$ is $225 \mathrm{m}$. Find an expression in terms of $t$ for the total distance travelled by the vehicle when $t⩾15 \mathrm{s}$.

A vehicle is moving in a straight line. The velocity $v \mathrm{m}{ \mathrm{s}}^{-1}$ at time $t \mathrm{s}$ after the vehicle starts is given by $v=A\left(t-0.05t^2\right)$ for $0 \mathrm{s}⩽t⩽15 \mathrm{s}$ and $v=\frac{B}{t^2}$ for $t⩾15 \mathrm{s}$, where $A$ and $B$ are constants. The distance travelled by the vehicle between $t=0 \mathrm{s}$ and $t=15 \mathrm{s}$ is $225 \mathrm{m}$.

Find the speed of the vehicle when it has travelled a total distance of $315 \mathrm{m}$.

Two walkers, $P$ and $Q$, travel along a straight track $ABC$. Both walkers start from point $A$ at time $t=0 \mathrm{s}$ and pass through point $B$ at time $t=10 \mathrm{s}$. They both finish at point $C$. $P$ starts from point $A$ with speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$ and accelerates with constant acceleration $0.1 \mathrm{m}{ \mathrm{s}}^{-2}$ until reaching point $B$.

Show that the distance from $A$ to $B$ is $25 \mathrm{m}$.

Two walkers, $P$ and $Q$, travel along a straight track $ABC$. Both walkers start from point $A$ at time $t=0 \mathrm{s}$ and pass through point $B$ at time $t=10 \mathrm{s}$. They both finish at point $C$. $P$ starts from point $A$ with speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$ and accelerates with constant acceleration $0.1 \mathrm{m}{ \mathrm{s}}^{-2}$ until reaching point $B$.

Find the speed of $P$ on reaching point $B$.

Two walkers, $P$ and $Q$, travel along a straight track $ABC$. Both walkers start from point $A$ at time $t=0 \mathrm{s}$ and pass through point $B$ at time $t=10 \mathrm{s}$. They both finish at point $C$. $P$ starts from point $A$ with speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$ and accelerates with constant acceleration $0.1 \mathrm{m}{ \mathrm{s}}^{-2}$ until reaching point $B$. $Q$ starts from point $A$ and moves with speed $v_1 \mathrm{m}{ \mathrm{s}}^{-1}$, given by $v_1=0.003t^2+0.06t+k$. When $Q$ passes through point $B$ both walkers have the same speed. Find the value of the constant $k$.

Two walkers, $P$ and $Q$, travel along a straight track $ABC$. Both walkers start from point $A$ at time $t=0 \mathrm{s}$ and pass through point $B$ at time $t=10 \mathrm{s}$. They both finish at point $C$. $P$ starts from point $A$ with speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$ and accelerates with constant acceleration $0.1 \mathrm{m}{ \mathrm{s}}^{-2}$ until reaching point $B$.

$Q$ starts from point $A$ and moves with speed $v_1 \mathrm{m}{ \mathrm{s}}^{-1}$, given by $v_1=0.003t^2+0.06t+k$. When $Q$ passes through point $B$ both walkers have the same speed.

$P$ moves from point $B$ to point $C$ with speed $v_2 \mathrm{m}{ \mathrm{s}}^{-1}$, given by $v_2=4-0.1t$, and comes to rest as $C$ is reached.

Show that the distance from $A$ to $C$ is $70 \mathrm{m}$.

Two walkers, $P$ and $Q$, travel along a straight track $ABC$. Both walkers start from point $A$ at time $t=0 \mathrm{s}$ and pass through point $B$ at time $t=10 \mathrm{s}$. They both finish at point $C$. $P$ starts from point $A$ with speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$ and accelerates with constant acceleration $0.1 \mathrm{m}{ \mathrm{s}}^{-2}$ until reaching point $B$. $Q$ starts from point $A$ and moves with speed $v_1 \mathrm{m}{ \mathrm{s}}^{-1}$, given by $v_1=0.003t^2+0.06t+k$. When $Q$ passes through point $B$ both walkers have the same speed. $P$ moves from point $B$ to point $C$ with speed $v_2 \mathrm{m}{ \mathrm{s}}^{-1}$, given by $v_2=4-0.1t$, and comes to rest as $C$ is reached. $Q$ moves from point $B$ to point $C$ with speed $v_3 \mathrm{m}{ \mathrm{s}}^{-1}$, given by $v_3=0.4t-0.01t^2$. Show that $Q$ reaches point $C$ first.