Pendulum and Spring Collision Calculator: Impact Speed, Restitution, Energy Lost and Spring Compression

Follow a pendulum from release, through a collision with a block, to the compression of a spring.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the mass, length and release angle of the pendulum, the mass of the block, the coefficient of restitution, the distance to the spring, the floor friction and the spring stiffness.

The calculator finds the speed of the pendulum at the bottom from energy conservation, then the speeds after the collision from momentum conservation and the coefficient of restitution.

It gives the energy lost, how high the pendulum rebounds, the speed of the block when it reaches the spring, the greatest compression and force of the spring and the time on the spring.

Formulas used

  • v_1 = \sqrt{2gl(1 - \cos\varphi_0)}
  • v_2 = \frac{(1 + e)m_1v_1}{m_1 + m_2}
  • v_1' = \frac{(m_1 - e m_2)v_1}{m_1 + m_2}
  • \Delta x = v_3\sqrt{m_2/k}

Worked example

With the default numbers in the calculator above:

  1. Height the pendulum falls: 0.75 m
  2. Speed of the pendulum at the bottom, just before the hit: 3.84 m/s
  3. Speed of the block just after the hit (momentum conserved, with restitution e): 4.60 m/s
  4. Speed of the pendulum just after the hit: 1.53 m/s
  5. Energy lost in the collision: 1.77 J (12.00 percent of the energy before the hit)
  6. Height the pendulum rises after the hit, and the angle it reaches: 0.12 m, which is 23.07 degrees from the vertical
  7. Speed of the block when it reaches the spring: 4.60 m/s
  8. Greatest compression of the spring: 102.93 mm
  9. Greatest spring force: 205.86 N
  10. Time the block is on the spring: 0.0702 s (half a cycle)

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Coefficient of restitution (1 = elastic, 0 = the two stick together) 1.

  1. Height the pendulum falls: 0.75 m
  2. Speed of the pendulum at the bottom, just before the hit: 3.84 m/s
  3. Speed of the block just after the hit (momentum conserved, with restitution e): 5.11 m/s
  4. Speed of the pendulum just after the hit: 1.28 m/s
  5. Energy lost in the collision: 0.00 J (0.00 percent of the energy before the hit)
  6. Height the pendulum rises after the hit, and the angle it reaches: 0.0833 m, which is 19.19 degrees from the vertical
  7. Speed of the block when it reaches the spring: 5.11 m/s
  8. Greatest compression of the spring: 114.37 mm
  9. Greatest spring force: 228.74 N
  10. Time the block is on the spring: 0.0702 s (half a cycle)

Common questions

What is the coefficient of restitution?

The ratio of the speed of separation to the speed of approach. 1 is perfectly elastic and 0 means the two stick together.

Is energy conserved in the collision?

Only if e is 1. Momentum is conserved in every collision with no outside force.

How is the spring compression found?

All the kinetic energy of the block goes into the spring, so one half m v squared equals one half k x squared.

What if the pendulum comes to a stop?

When m1 equals e times m2 the pendulum is left at rest after the hit and passes all its motion to the block.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any university physics text (collisions and the coefficient of restitution); Meriam and Kraige, Engineering Mechanics: Dynamics.

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