M1 June 2014 (R) Q3

EdexcelOld spec13 marksSUVAT Time-Series Graphs

3. A car starts from rest and moves with constant acceleration along a straight horizontal road. The car reaches a speed of \(V\) m s\(^{-1}\) in 20 seconds. It moves at constant speed \(V\) m s\(^{-1}\) for the next 30 seconds, then moves with constant deceleration \(\dfrac{1}{2}\) m s\(^{-2}\) until it has speed 8 m s\(^{-1}\). It moves at speed 8 m s\(^{-1}\) for the next 15 seconds and then moves with constant deceleration \(\dfrac{1}{3}\) m s\(^{-2}\) until it comes to rest.

(a) Sketch, in the space below, a speed-time graph for this journey. (3)

In the first 20 seconds of this journey the car travels 140 m.

Find

(b) the value of \(V\), (2)

(d) the total distance travelled by the car. (4)

Speed-time graph: rises from 0 to V over 20 s (area 140 m), constant V for 30 s, falls to 8 over t1, constant 8 for 15 s, falls to 0 over t2 — Mark scheme (a)

Scheme	Marks
\(0 < t < 50\)	B1
\(50 < t\)	B1
(\(V\),8,15,20,30)	B1
	(3)

Mark scheme (b)

Scheme	Marks
Use area under graph or suvat to form an equation in \(V\) only.
\(140 = \dfrac{1}{2} \times 20 \times V\)	M1
\(V = 14\)	A1
	(2)

Notes

M1 for use of area under graph (must have ‘1/2’) or suvat to obtain an equation in \(V\) only.

A1 for \(V = 14\)

Mark scheme (c)

Scheme	Marks
\(8 = V - \dfrac{1}{2}t_1\) (and /or \(0 = 8 - \dfrac{1}{3}t_2\))	M1
\(t_1 = 12\), (and/or \(t_2 = 24\))	A1
Total time \(= 20 + 30 + t_1 + 15 + t_2 = 101\) (seconds)	DM1 A1
	(4)

Notes

First M1 for use of either \(8 = V - \frac{1}{2}t_1\) or \(0 = 8 - \frac{1}{3}t_2\)

First A1 for either \(t_1 = 12\) or \(t_2 = 24\)

Second M1, dependent on the first M1, for \(20 + 30 + t_1 + 15 + t_2\) (must include all 5 times)

Second A1 for 101 (s)

Mark scheme (d)

Scheme	Marks
Total distance \(= 140 + 30V + \dfrac{V + 8}{2}t_1 + 15 \times 8 + \dfrac{1}{2} \times 8 \times t_2\)	M1A2 ft
\(= 140 + 30 \times 14 + 11 \times 12 + 15 \times 8 + 24 \times 4\)
\(= 908\) (m)	A1
	(4)
	(13 marks)

Notes

First M1 for an expression for the total area (distance) including all parts of the motion. Where a triangle or trapezium is used, a ‘1/2’ must be seen.

Second A2 ft on their \(V\), \(t_1\) and \(t_2\), -1 each error.

Fourth A1 for 908 (m).