Edexcel A-Level Physics A2 4.3 Further Mechanics Questions

Practise Edexcel A-Level Physics further mechanics through impulse, two-dimensional collisions, kinetic energy and circular-motion forces.

Exam points

calculate impulse or mean force from vector changes in momentum during collisions
test elastic collisions using momentum components and total kinetic-energy comparisons
apply angular velocity and centripetal-force equations to circular-motion contexts

Covered syllabus points

4.3.81 - Impulse and change of momentum 4.3.82 - Core Practical 9 - force and momentum change 4.3.83 - Momentum conservation in two dimensions 4.3.84 - Core Practical 10 - ICT collision analysis 4.3.85 - Elastic and inelastic collisions 4.3.86 - Kinetic energy from momentum 4.3.87 - Angular displacement 4.3.88 - Angular velocity 4.3.89 - Centripetal acceleration derivation 4.3.90 - Centripetal force requirement 4.3.91 - Centripetal force equations

[Maximum number: 8]

Some asteroids pass very close to the Earth. Scientists are planning methods to deflect asteroids, to prevent them hitting the Earth.

One method would involve colliding a spacecraft into the surface of the asteroid, to change the path and speed of the asteroid. The spacecraft would remain joined to the asteroid after the collision.

(a)

This collision method is modelled for a spacecraft travelling in a direction at $90^{\circ}$ to the path of the asteroid.

Sketch a labelled vector diagram to show the momenta of the bodies before and after the collision.

[ 2 ]

(b)

Show that this collision method causes the asteroid to change its direction through an angle of about $10^{-7}$ radian.

Momentum of asteroid = $7.6\times10^{13}\,\mathrm{N\,s}$

[ 2 ]

Use trigonometry with the mutually perpendicular momenta.

$\tan\theta=\frac{1.1\times10^7}{7.6\times10^{13}}=1.45\times10^{-7}$

Hence $\theta=1.45\times10^{-7}\,\mathrm{rad}$ , approximately $1.5\times10^{-7}\,\mathrm{rad}$ .

(c)

After the collision, the asteroid and spacecraft remain joined and move together.

Calculate the component of their velocity at 90° to the original path of the asteroid after the collision.
mass of asteroid = 2.8 × 10⁹ kg

Component of velocity =

[ 2 ]

(d)

Another method would involve attaching a rocket motor to the asteroid and using the motor to apply a force to the asteroid.

In this method the force is applied at $90^\circ$ to the path of the asteroid. Deduce whether this would produce a change in momentum as great as the change produced by the collision method.

Force exerted by rocket motor = $5.1\times10^6\,\mathrm{N}$

Time for which rocket motor applies force = $6\,\mathrm{minutes}$

[ 2 ]

Use impulse $\Delta p=F\Delta t$ .

$\Delta p=(5.1\times10^6\,\mathrm{N})(6\times60\,\mathrm{s})=1.8\times10^9\,\mathrm{N\,s}$

This is greater than the $1.1\times10^7\,\mathrm{N\,s}$ change from the collision method.

[Maximum number: 12]

Hockey is a sport played with a stick and a ball. The player tries to hit the ball with part of the stick called the 'heel', as shown.

Figure for Question 2 — Edexcel A-Level Physics A2

(a)

The player swings her stick so that the heel moves horizontally in a circle of radius 0.80 m across the ground, as shown below.

Figure for Question (a) — Edexcel A-Level Physics A2

It takes a time of 0.22 s for the heel to move through an angle of 1.3 radians.
Calculate the speed of the heel.

[ 3 ]

The angular speed is $\omega=\theta/t=1.3/0.22=5.9\,\mathrm{rad\,s^{-1}}$ . Hence $v=r\omega=(0.80)(5.9)=4.73\,\mathrm{m\,s^{-1}}.$

(b)

A ball has a speed of $13 \mathrm{~ms}^{-1}$ . The heel has a momentum $p_{\text {heel }}$ of 3.0 Ns , in the direction of the dashed line, as shown.

The heel collides with the ball and stops.
The speed of the ball after being hit by the heel is $16 \mathrm{~ms}^{-1}$ .

Figure for Question (b) — Edexcel A-Level Physics A2

[ 9 ]

(i)

Deduce whether momentum is conserved for this collision by completing the vector diagram below. A scaled line representing $p_{\text {heel }}$ is shown.
mass of ball $=160 \mathrm{~g}$

[ 5 ]

The ball momenta are $p_i=(0.160)(13)=2.08\,\mathrm{N\,s}$ and $p_f=(0.160)(16)=2.56\,\mathrm{N\,s}$ . Using the stated scale, draw these vectors with the given heel momentum $3.0\,\mathrm{N\,s}$ . Momentum is conserved if the three vectors form a closed triangle.

Figure for Answer to Question (i) — Edexcel A-Level Physics A2

(ii)

Deduce whether the collision is elastic.
speed of heel before collision = 5.0 m s^-1

[ 4 ]

Before collision, $E_k=\tfrac12p_{\rm heel}v_{\rm heel}+\tfrac12m v_i^2=7.5+13.5=21.0\,\mathrm J$ . After collision, $E_k=\tfrac12(0.160)(16)^2=20.5\,\mathrm J$ . The values are not equal, so the collision is not elastic.

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question 2

Question (a)

Question (b)

Question (i)

Question (ii)