M1: Kinematics of a particle moving in a straight lineEdexcel International A Level Maths: Topic test

1. 1
   A skier moves down a straight slope with constant acceleration. She passes a point $A$ with speed $3$ m s⁻¹ and passes a point $B$, which is $40$ m further down the slope, with speed $11$ m s⁻¹.

   (a)

   Find the acceleration of the skier.

   [1 mark]

   • A$0.7$ m s⁻²
   • B$2.8$ m s⁻²
   • C$1.4$ m s⁻²
   • D$0.2$ m s⁻²

   (b)

   Find the time taken to travel from $A$ to $B$.

   [1 mark]

   • A$5.7$ s
   • B$2.9$ s
   • C$3.6$ s
   • D$13.3$ s

   (c)

   Find the distance from $A$ at which the speed of the skier is $7$ m s⁻¹.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The speed of a toy train along a straight track is modelled as follows. It increases uniformly from rest to $0.8$ m s⁻¹ in $5$ s, then stays constant for $12$ s, and then decreases uniformly to rest in $4$ s.

   (a)

   Find the total distance travelled by the train.

   [1 mark]

   • A$9.6$ m
   • B$13.2$ m
   • C$16.8$ m
   • D$8.4$ m

   (b)

   Find the magnitude of the deceleration in the final section.

   [1 mark]

   • A$0.16$ m s⁻²
   • B$0.067$ m s⁻²
   • C$5$ m s⁻²
   • D$0.2$ m s⁻²

   (c)

   Find the time, measured from the start, at which the speed of the train is $0.5$ m s⁻¹ while it is slowing down.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A stone is dropped from rest from the top of a vertical cliff of height $78.4$ m above the sea. Model the stone as a particle moving freely under gravity and take $g=9.8$ m s⁻².

   (a)

   Find the time taken for the stone to reach the sea.

   [3 marks]

   (b)

   A second stone is thrown vertically downwards from the top of the cliff $1$ s after the first stone is dropped. The two stones reach the sea at the same instant. Find the speed with which the second stone is thrown.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two drones, $A$ and $B$, fly along the same straight horizontal line. At time $t=0$, drone $A$ passes a point $O$ with constant speed $8$ m s⁻¹. At the same instant drone $B$ starts from rest at $O$ and accelerates uniformly at $0.5$ m s⁻² until it reaches a speed of $12$ m s⁻¹, after which it flies at that constant speed.

   (a)

   Show that $B$ has not overtaken $A$ at the instant $B$ reaches a speed of $12$ m s⁻¹, and find the distance between the drones at that instant.

   [6 marks]

   (b)

   Using the areas under the speed–time graphs of the two drones, or otherwise, find the time at which $B$ overtakes $A$, and the distance of the drones from $O$ at that instant.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A robot moves along a straight rail. At $t=0$ it leaves a point $O$ and moves away from $O$ with constant velocity $0.4$ m s⁻¹ for $15$ s. It then remains at rest for $10$ s, and then returns to $O$ with constant velocity, arriving at $O$ at $t=55$ s.

   (a)

   Find the distance of the robot from $O$ at $t=20$.

   [1 mark]

   • A$8$ m
   • B$0$ m
   • C$0.4$ m
   • D$6$ m

   (b)

   Taking the direction away from $O$ as positive, find the velocity of the robot while it returns to $O$.

   [1 mark]

   • A$0.2$ m s⁻¹
   • B$-0.2$ m s⁻¹
   • C$-0.11$ m s⁻¹
   • D$-0.4$ m s⁻¹

   (c)

   Find the average speed of the robot over the whole journey.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A tram travelling at $18$ m s⁻¹ brakes with constant deceleration and comes to rest after travelling $90$ m.

   (a)

   Find the magnitude of the deceleration of the tram.

   [1 mark]

   • A$1.8$ m s⁻²
   • B$3.6$ m s⁻²
   • C$0.9$ m s⁻²
   • D$0.2$ m s⁻²

   (b)

   Find the time taken for the tram to stop.

   [1 mark]

   • A$5$ s
   • B$7.1$ s
   • C$10$ s
   • D$50$ s

   (c)

   Find the speed of the tram when it has travelled $40$ m from the point where it began to brake.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle moves along a straight line with constant acceleration $a$ m s⁻². At time $t=0$ it passes a point $O$ with speed $u$ m s⁻¹. At $t=4$ s the particle is $36$ m from $O$ and at $t=8$ s it is $112$ m from $O$, both distances being measured in the direction of motion.

   (a)

   Find the values of $u$ and $a$.

   [3 marks]

   (b)

   Find the distance travelled by the particle during the eighth second, that is between $t=7$ and $t=8$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A toy car moves along a straight track. At $t=0$ it passes a point $O$ with velocity $6$ m s⁻¹ in the positive direction and for the first $8$ s it has constant acceleration $-1.5$ m s⁻². From $t=8$ s the car moves with constant velocity until time $T$ s, and then decelerates uniformly at $2$ m s⁻² until it comes to rest at a point $45$ m from $O$ on the negative side of $O$.

   (a)

   Find the displacement of the car from $O$ at $t=8$, and the total distance travelled by the car during the first $8$ s.

   [6 marks]

   (b)

   Find the value of $T$.

   [6 marks]

   Total for question 8: 12 marks

End of questions