M1: Dynamics of a particle moving in a straight line or planeEdexcel International A Level Maths: Topic test

1. 1
   A sledge of mass $40$ kg is pulled from rest across horizontal ice by a horizontal rope. The tension in the rope is $120$ N and the total resistance to the motion of the sledge is a constant $20$ N.

   (a)

   Find the acceleration of the sledge.

   [1 mark]

   • A$3.0$ m s⁻²
   • B$3.5$ m s⁻²
   • C$2.5$ m s⁻²
   • D$0.5$ m s⁻²

   (b)

   Find the distance travelled by the sledge in the first $6$ s.

   [1 mark]

   • A$45$ m
   • B$90$ m
   • C$54$ m
   • D$15$ m

   (c)

   The rope snaps after $6$ s. Find the time, from the instant the rope snaps, for which the sledge continues to move.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A railway truck $A$ of mass $800$ kg moves at $3$ m s⁻¹ along a smooth straight horizontal track and collides directly with a stationary truck $B$ of mass $1200$ kg. The two trucks couple together and move on as one body.

   (a)

   Find the speed of the trucks immediately after the collision.

   [1 mark]

   • A$1.8$ m s⁻¹
   • B$1.2$ m s⁻¹
   • C$1.5$ m s⁻¹
   • D$2$ m s⁻¹

   (b)

   Find the magnitude of the impulse exerted by $A$ on $B$ in the collision.

   [1 mark]

   • A$960$ N s
   • B$2400$ N s
   • C$1800$ N s
   • D$1440$ N s

   (c)

   The collision lasts $0.4$ s. Find the magnitude of the average force exerted by $A$ on $B$ during the collision.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A suitcase of mass $25$ kg is pushed along a rough horizontal floor by a constant horizontal force of $100$ N. The coefficient of friction between the suitcase and the floor is $0.3$. Take $g=9.8$ m s⁻².

   (a)

   Find the acceleration of the suitcase.

   [3 marks]

   (b)

   The suitcase is moving at $4$ m s⁻¹ when the pushing force is removed. Find the distance it travels before coming to rest.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $A$ of mass $4$ kg rests on a rough horizontal table. The coefficient of friction between $A$ and the table is $0.25$. $A$ is attached to one end of a light inextensible string. The string passes over a smooth light pulley fixed at the edge of the table, and a particle $B$ of mass $6$ kg hangs freely from the other end, $1.5$ m above the floor. The string between $A$ and the pulley is horizontal and the system is released from rest with the string taut. The table is long enough for $A$ not to reach the pulley. Take $g=9.8$ m s⁻².

   (a)

   Find the acceleration of the particles and the tension in the string before $B$ reaches the floor.

   [6 marks]

   (b)

   Find the total distance $A$ moves from the instant of release until it comes to rest.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A toboggan of mass $5$ kg is released from rest on a smooth snow slope inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha=\frac35$. Model the toboggan as a particle and take $g=9.8$ m s⁻².

   (a)

   Find the acceleration of the toboggan down the slope.

   [1 mark]

   • A$7.84$ m s⁻²
   • B$9.8$ m s⁻²
   • C$4.9$ m s⁻²
   • D$5.88$ m s⁻²

   (b)

   Find the speed of the toboggan after it has travelled $2.5$ m down the slope.

   [1 mark]

   • A$29.4$ m s⁻¹
   • B$5.42$ m s⁻¹
   • C$6.26$ m s⁻¹
   • D$7.00$ m s⁻¹

   (c)

   Find the magnitude of the normal reaction of the slope on the toboggan.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A footballer kicks a stationary football of mass $0.45$ kg. The ball leaves the boot with a speed of $20$ m s⁻¹ and the boot is in contact with the ball for $0.015$ s. Model the force of the boot on the ball as constant and ignore the weight of the ball during the kick.

   (a)

   Find the magnitude of the impulse of the boot on the ball.

   [1 mark]

   • A$9$ N s
   • B$0.135$ N s
   • C$20$ N s
   • D$600$ N s

   (b)

   Find the magnitude of the average force exerted by the boot on the ball.

   [1 mark]

   • A$0.135$ N
   • B$9$ N
   • C$600$ N
   • D$300$ N

   (c)

   A second stationary ball of mass $0.5$ kg is given an impulse of the same magnitude by the boot. Find the speed with which it leaves the boot.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A car of mass $1200$ kg pulls a trailer of mass $300$ kg along a straight horizontal road by means of a light horizontal tow bar. The driving force of the car's engine is $3000$ N. The resistance to the motion of the car is $400$ N and the resistance to the motion of the trailer is $200$ N.

   (a)

   Find the acceleration of the car and trailer.

   [3 marks]

   (b)

   The car and trailer are moving along the road when the engine is switched off. Assuming the resistances are unchanged, find the tension in the tow bar immediately afterwards.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle $A$ of mass $2$ kg moves at $9$ m s⁻¹ along a straight line on a horizontal surface and collides directly with a stationary particle $B$ of mass $3$ kg. After the collision $A$ and $B$ move together as a single combined particle. The collision takes place on a smooth part of the surface. The combined particle then slides onto a rough part of the surface, where the coefficient of friction is $0.25$, and comes to rest. Take $g=9.8$ m s⁻².

   (a)

   Find the speed of the combined particle immediately after the collision, and the magnitude of the impulse exerted by $A$ on $B$.

   [6 marks]

   (b)

   Find the distance, and the time, for which the combined particle slides on the rough surface before it comes to rest.

   [6 marks]

   Total for question 8: 12 marks

End of questions