Focus: Distance-Time & Speed-Time Graphs, Acceleration, Area as Distance & Average Speed
A car travels from Town $P$ to Town $Q$. The distance-time graph shows the journey where the car travels $80\text{ km}$ in the first 1 hour, stops for 30 minutes at a rest area, and travels the remaining $60\text{ km}$ in 45 minutes.
(a) State the duration, in minutes, that the car was stationary. [1 mark]
(b) Calculate the speed, in $\text{km/h}$, of the car during the final 45 minutes. [2 marks]
(c) Calculate the average speed, in $\text{km/h}$, for the entire journey. [1 mark]
(a) Stationary time
(b) Speed = $\frac{60\text{ km}}{45/60\text{ h}} = \frac{60}{0.75}$
(c) Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}} = \frac{80 + 60}{1 + 0.5 + 0.75} = \frac{140}{2.25}$
The speed of an electric train increases uniformly from $15\text{ m/s}$ to $35\text{ m/s}$ in $8\text{ seconds}$, then maintains constant speed for $12\text{ seconds}$, and finally decelerates to a stop in $10\text{ seconds}$.
(a) Calculate the rate of change of speed (acceleration) during the first 8 seconds. [2 marks]
(b) Calculate the magnitude of deceleration during the final 10 seconds. [2 marks]
(a) Acceleration $a = \frac{35 - 15}{8} = \frac{20}{8}$
(b) Deceleration $= \frac{0 - 35}{10} = -3.5\text{ m/s}^2$
In a speed-time graph, a particle moves from rest to a speed of $v\text{ m/s}$ in $5\text{ seconds}$. The area under the graph for the first 5 seconds is $75\text{ m}$. Calculate the value of $v$.
Area of triangle $= \frac{1}{2} \times \text{base} \times \text{height} = 75$
Solve for $v$
The speed-time graph shows the motion of an express bus for a period of $T\text{ seconds}$. The bus starts from rest, accelerates uniformly to $24\text{ m/s}$ in $10\text{ seconds}$, cruises at this speed for $15\text{ seconds}$, and then decelerates to a stop in $(T - 25)\text{ seconds}$. The total distance travelled is $540\text{ m}$.
(a) State the uniform speed of the bus. [1 mark]
(b) Calculate the value of $T$. [4 marks]
(c) Calculate the deceleration of the bus during the final $(T - 25)\text{ seconds}$. [2 marks]
(d) Find the average speed of the bus for the whole journey. [2 marks]
(a) Uniform speed
(b) Area of trapezium $= \frac{1}{2}(15 + T) \times 24 = 540$
(c) Final stage time $= 30 - 25 = 5\text{ s}$. Deceleration:
(d) Average speed
Motorcyclist $A$ and Cyclist $B$ start simultaneously from the same point along a straight track. Motorcyclist $A$ moves with uniform acceleration from rest to $30\text{ m/s}$ in $12\text{ seconds}$ and maintains that speed. Cyclist $B$ travels with a constant uniform speed of $18\text{ m/s}$.
(a) Find the distance travelled by Motorcyclist $A$ during the first 12 seconds. [2 marks]
(b) Calculate the distance between Motorcyclist $A$ and Cyclist $B$ at $t = 12\text{ seconds}$. [3 marks]
(c) Calculate the time $t$, in seconds, when Motorcyclist $A$ overtakes Cyclist $B$. [4 marks]
(a) Distance of $A$ at 12s: Area of triangle $= \frac{1}{2}(12)(30)$
(b) Distance of $B$ at 12s $= 18 \times 12 = 216\text{ m}$.
(c) Let time after 12s be $t_1$. At overtake: $180 + 30t_1 = 18(12 + t_1)$