Impulse in Collisions

Introduction to Impulse

• Definition: Impulse is the change in momentum of an object. It is a vector quantity.
• Symbol: $I$
• Units: Newton-seconds (Ns) or kg m/s
• Formula: $I = \Delta p = m\Delta v = F\Delta t$
  • $I$ = Impulse
  • $\Delta p$ = Change in momentum
  • $m$ = Mass
  • $\Delta v$ = Change in velocity
  • $F$ = Force
  • $\Delta t$ = Time interval

KEY TAKEAWAY: Impulse is the product of the force acting on an object and the time interval over which it acts, resulting in a change in the object’s momentum.

Impulse and Momentum Theorem

• The impulse-momentum theorem states that the impulse acting on an object is equal to the change in momentum of that object.
• Equation: $F\Delta t = m\Delta v$
• This equation is derived directly from Newton’s second law of motion:
  • $F = ma$
  • $a = \frac{\Delta v}{\Delta t}$
  • $F = m\frac{\Delta v}{\Delta t}$
  • $F\Delta t = m\Delta v$

VCAA FOCUS: Be prepared to apply the impulse-momentum theorem to solve problems involving collisions and changes in motion.

Isolated Systems and Collisions

• Isolated System: A system where the net external force acting on it is zero.
• Collision: An event where two or more objects exert forces on each other for a relatively short period.
• In an isolated system: The total momentum before a collision equals the total momentum after the collision (law of conservation of momentum).

Conservation of Momentum

• Law of Conservation of Momentum: In an isolated system, the total momentum remains constant.
• Equation: $\sum p_{initial} = \sum p_{final}$
  • $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$
  • Where:
    • $m_1$ and $m_2$ are the masses of the objects.
    • $v_{1i}$ and $v_{2i}$ are the initial velocities of the objects.
    • $v_{1f}$ and $v_{2f}$ are the final velocities of the objects.

EXAM TIP: Always define a positive direction when dealing with momentum and impulse questions.

Types of Collisions

• Elastic Collision: A collision in which total kinetic energy is conserved.
  • Kinetic energy before = Kinetic energy after
  • $KE_{initial} = KE_{final}$
  • $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$
• Inelastic Collision: A collision in which total kinetic energy is not conserved. Some kinetic energy is converted into other forms of energy (e.g., heat, sound, deformation).
  • Kinetic energy before ≠ Kinetic energy after
  • $KE_{initial} \neq KE_{final}$

COMMON MISTAKE: Confusing conservation of momentum with conservation of kinetic energy. Momentum is always conserved in an isolated system, but kinetic energy is only conserved in elastic collisions.

Impulse in Straight Line Collisions (1D)

• Consider two objects moving along a straight line (one dimension) colliding.
• Apply the conservation of momentum principle: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$
• The impulse on object 1 is $I_1 = m_1(v_{1f} - v_{1i})$
• The impulse on object 2 is $I_2 = m_2(v_{2f} - v_{2i})$
• According to Newton’s Third Law, the force exerted by object 1 on object 2 is equal and opposite to the force exerted by object 2 on object 1. Therefore, the impulses are equal in magnitude and opposite in direction: $I_1 = -I_2$

Practical Investigations and Analysis

• Experimental Setup: Use trolleys on a track with sensors to measure velocities before and after collisions.
• Procedure:
  1. Measure the masses of the trolleys.
  2. Record the initial and final velocities of the trolleys using motion sensors.
  3. Calculate the initial and final momentum of each trolley.
  4. Calculate the total momentum before and after the collision.
  5. Determine the impulse experienced by each trolley.
  6. Calculate the change in kinetic energy to determine if the collision is elastic or inelastic.
• Analysis:
  • Compare the total momentum before and after the collision to verify conservation of momentum.
  • Calculate the percentage difference to account for experimental errors.
  • Analyze the change in kinetic energy to classify the collision type.
• Sources of Error:
  • Friction between the trolleys and the track.
  • Inaccurate velocity measurements.
  • External forces not accounted for.

STUDY HINT: Practice solving a variety of impulse and collision problems, including both elastic and inelastic collisions. Pay attention to the signs of velocities and impulses.

Examples

Example 1: Inelastic Collision

A 2.0 kg trolley moving at 3.0 m/s collides with a stationary 1.0 kg trolley. After the collision, the 2.0 kg trolley moves at 1.0 m/s in the same direction.

1. What is the velocity of the 1.0 kg trolley after the collision?
2. What is the impulse on the 2.0 kg trolley?

Solution:

1. Conservation of momentum:
   • $(2.0 kg)(3.0 m/s) + (1.0 kg)(0 m/s) = (2.0 kg)(1.0 m/s) + (1.0 kg)v_{2f}$
   • \$6.0 = 2.0 + v_{2f}$
   • $v_{2f} = 4.0 m/s$
2. Impulse on the 2.0 kg trolley:
   • $I = m\Delta v = (2.0 kg)(1.0 m/s - 3.0 m/s) = -4.0 Ns$

Example 2: Elastic Collision

A 5.0 kg bowling ball moving at 8.0 m/s strikes a stationary 1.0 kg bowling pin. After the collision, the bowling pin moves at 12.0 m/s. Assuming the collision is perfectly elastic, calculate the final velocity of the bowling ball.

Solution:

1. Conservation of momentum:
   • $(5.0 kg)(8.0 m/s) + (1.0 kg)(0 m/s) = (5.0 kg)v_{1f} + (1.0 kg)(12.0 m/s)$
   • \$40 = 5v_{1f} + 12$
   • $5v_{1f} = 28$
   • $v_{1f} = 5.6 m/s$

REMEMBER: Impulse can be positive or negative, indicating the direction of the force and change in momentum.

Force-Time Graphs and Impulse

• The impulse can also be found by calculating the area under a force-time graph.
• Area under F-t graph = Impulse
• For a constant force, the area is simply a rectangle: $Area = F\Delta t = Impulse$
• For a variable force, the area can be found using integration or by approximating the area with smaller shapes (e.g., triangles, rectangles).

APPLICATION: Understanding impulse is crucial in designing safety equipment such as car airbags and crumple zones, which increase the time over which the force acts during a collision, reducing the force experienced by the occupants.