Further Mechanics: Momentum, Impulse and Collisions Depth
Momentum, Impulse and Collisions Depth, a Further Mechanics topic, extends direct (one-dimensional) collision analysis to oblique impact, where a smooth sphere strikes a fixed plane surface at an angle rather than square-on. Because the surface is smooth, it can exert no force along its own surface, so the component of the sphere's velocity parallel to the surface is unchanged by the impact; only the component perpendicular to the surface is affected, and it obeys Newton's law of restitution exactly as in direct impact. The topic also covers finding the kinetic energy lost in a collision (direct or oblique), and working through successive collisions between several bodies in the correct time order, using the outcome of one collision as the input to the next.
Before you start
Make sure you're comfortable with these topics first:
Method
- For oblique impact of a smooth sphere with a fixed plane, resolve the velocity into two components: one perpendicular (normal) to the plane, and one parallel (tangential) to it.
- The parallel component of velocity is unchanged by the impact, since a smooth surface exerts no force along its own surface.
- The perpendicular component obeys Newton's law of restitution exactly as in direct impact: rebound perpendicular speed = e x approach perpendicular speed, with its direction reversed.
- Recombine the unchanged parallel component and the rebounded perpendicular component (they are at right angles) to find the resultant speed and direction after impact, using Pythagoras' theorem and trigonometry.
- To find the kinetic energy lost in any collision, compute total KE = sum of (1/2)mv^2 for every body involved, before and after, and subtract; this loss can never be negative, since 0 <= e <= 1 guarantees energy is not created.
- For an impulse in an oblique impact, use impulse = m(v-u) as a vector, and note the impulse acts entirely along the perpendicular (normal) direction, since the parallel component of velocity does not change.
- For successive or multi-body collisions, work through them strictly in time order: find the velocities after the first collision, then use the relevant one as the initial (u) velocity for the next collision.
Worked example
A smooth sphere strikes a fixed smooth wall with speed 10 m/s, at an angle of 30 degrees to the wall. The coefficient of restitution between the sphere and the wall is 0.6. Find the speed and direction of the sphere immediately after impact.
- Resolve the initial velocity into components parallel and perpendicular to the wall: parallel = 10cos(30) = 5sqrt(3) m/s; perpendicular (toward the wall) = 10sin(30) = 5 m/s.
- The parallel component is unchanged by the impact, since the wall is smooth: parallel component after impact = 5sqrt(3) m/s, still directed along the wall the same way.
- The perpendicular component obeys Newton's law of restitution: rebound perpendicular speed = e x approach perpendicular speed = 0.6 x 5 = 3 m/s, now directed away from the wall.
- Combine the two components, which are at right angles: speed = sqrt((5sqrt(3))^2 + 3^2) = sqrt(75+9) = sqrt(84) = 2sqrt(21) m/s, approximately 9.17 m/s (3 s.f.).
- Find the angle to the wall after impact: tan(alpha) = perpendicular/parallel = 3/(5sqrt(3)) = sqrt(3)/5, giving alpha = arctan(sqrt(3)/5), approximately 19.1 degrees (3 s.f.).
Practice questions
Try each question, then tap to reveal the answer.
Q1State what happens to the component of velocity of a smooth sphere parallel to a fixed plane surface during an oblique impact, and explain why.Show answer
Answer: It is unchanged, because a smooth surface can exert no force, and so no impulse, along its own surface; it can only exert a force perpendicular to itself (along the normal).
Q2A smooth sphere hits a fixed wall with velocity components 8 m/s parallel to the wall and 6 m/s perpendicular to the wall (approaching). The coefficient of restitution is 0.5. Find the components of velocity immediately after impact.Show answer
Answer: Parallel component unchanged = 8 m/s; perpendicular component = 0.5 x 6 = 3 m/s, now directed away from the wall.
Q3For the sphere in the previous question, find its speed immediately after impact.Show answer
Answer: sqrt(8^2+3^2) = sqrt(64+9) = sqrt(73), approximately 8.54 m/s (3 s.f.).
Q4Two identical smooth spheres A and B, each of mass m, move toward each other on a line with speeds 5 m/s and 1 m/s respectively and collide directly with coefficient of restitution 0.8. Find the kinetic energy lost in the collision, in terms of m.Show answer
Answer: Taking A's direction as positive, momentum gives v_A+v_B=4 and restitution gives v_B-v_A=4.8, so v_A=-0.4 m/s and v_B=4.4 m/s. KE before = (1/2)m(25+1)=13m J; KE after = (1/2)m(0.16+19.36)=9.76m J; KE lost = 3.24m J.
Q5A ball moving at 12 m/s strikes a smooth fixed surface at 45 degrees to the surface, with coefficient of restitution 1/3. Find the perpendicular component of its velocity before and after impact.Show answer
Answer: Before: 12sin(45) = 6sqrt(2) m/s, approximately 8.49 m/s. After: (1/3)(6sqrt(2)) = 2sqrt(2) m/s, approximately 2.83 m/s.
Q6Explain why the kinetic energy lost in a direct collision between two particles is zero only when e=1.Show answer
Answer: The kinetic energy lost equals (1/2)(m1m2/(m1+m2))(1-e^2)(approach speed)^2; since the masses and the approach speed are positive in a genuine collision, this expression is zero only when 1-e^2=0, i.e. e=1 (a perfectly elastic collision), and is positive for any e<1.
Q7Identical smooth spheres A and B lie at rest in a line on a smooth table. A is projected toward B at 6 m/s and they collide directly with coefficient of restitution 0.5. Find the speeds of A and B immediately after the collision.Show answer
Answer: Momentum (equal masses): v_A+v_B=6. Restitution: v_B-v_A=0.5(6-0)=3. Solving gives v_A=1.5 m/s and v_B=4.5 m/s.
Q8Using the result of the previous question, determine whether sphere A will catch up with sphere B again, giving a reason.Show answer
Answer: No: since v_B=4.5 m/s is greater than v_A=1.5 m/s and both move the same way, B moves away from A faster than A travels, so the gap between them can only increase.
Exam-style questions
Written in the style of a A Level Further Maths exam paper, with a full mark scheme.
A smooth sphere is projected across a smooth horizontal floor and strikes a fixed vertical wall with speed 8 m/s at an angle of 60 degrees to the wall. The coefficient of restitution between the sphere and the wall is 0.5. (a) Find the speed and the angle to the wall of the sphere immediately after this impact. (b) Given the sphere has mass 0.4 kg, find the kinetic energy lost in the impact.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 8 available
Three smooth spheres A, B and C, of mass 2 kg, 3 kg and 5 kg respectively, lie at rest in a straight line on a smooth horizontal table, in the order A, B, C. Sphere A is then projected toward B with speed 10 m/s. The coefficient of restitution between any two of the spheres is 0.5. (a) Find the velocities of A and B immediately after their collision. (b) Show that B then goes on to collide with C, and find the velocities of B and C immediately after this second collision. (c) Determine whether A and B collide again, giving a reason.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 10 available
See real A Level Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the further mechanics: momentum, impulse and collisions depth worksheet pack - 9 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
Next topics
Not quite what you needed?
Tell us what is missing on further mechanics: momentum, impulse and collisions depth, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.