SPH4U Momentum, Impulse & Collisions: Grade 12 Physics Practice Problems [Full Solutions]

In Ontario SPH4U Grade 12 Physics, linear momentum (p = mv) and impulse (J = FΔt = Δp) are conserved vector quantities in closed systems. Elastic collisions conserve both total momentum and kinetic energy (Ek), while inelastic collisions conserve momentum but convert kinetic energy into thermal energy and internal deformation.

Official Ontario Curriculum Breakdown: SPH4U

• Ministry Course Code: SPH4U (Physics, Grade 12, University Preparation)
• Prerequisite Course: SPH3U (Physics, Grade 11, University Preparation)
• Curriculum Strand: Dynamics (Forces, Momentum, Impulse, Conservation of Energy)
• Conservation Laws: Total initial momentum equals total final momentum (p_total_initial = p_total_final)
• Elastic Collision Criteria: Momentum conserved AND Kinetic Energy conserved (Ek_initial = Ek_final)

Worked Example 1: 1D Completely Inelastic Collision (Sticking Together)

Problem: A 1200 kg vehicle traveling north at 22 m/s collides on an icy road with an 1800 kg pickup truck traveling south at 14 m/s. If the vehicles lock bumpers upon impact, determine their combined final velocity immediately following the collision.

• Step 1 (Assign vector sign conventions): Designate North as positive (+) and South as negative (-).
• Step 2 (State conservation of momentum): m₁v₁ + m₂v₂ = (m₁ + m₂)v_final.
• Step 3 (Substitute given values): (1200)(+22) + (1800)(-14) = (1200 + 1800)v_final.
• Step 4 (Calculate initial momenta): 26,400 kg·m/s + (-25,200 kg·m/s) = 3000 v_final.
• Step 5 (Compute net momentum): +1200 kg·m/s = 3000 v_final.
• Step 6 (Solve for final velocity): v_final = 1200 / 3000 = +0.40 m/s.
• Step 7 (State final direction): The positive sign indicates North. The combined wreckage moves North at 0.40 m/s.

Teacher's Strategy: Always explicitly state your vector convention (+ and -) at the start of your solution. Momentum is a vector; treating velocities as positive scalars guarantees incorrect answers.

Worked Example 2: 2D Glancing Collision (Vector Components)

Problem: A 0.25 kg puck sliding east at 4.0 m/s across a frictionless air table collides with an identical stationary 0.25 kg puck. After impact, the first puck moves at 3.0 m/s at an angle of 30° north of east. Determine the final velocity (magnitude and direction) of the second puck.

• Step 1 (Resolve x-component momentum): Initial P_x = m₁(v₁_initial) = (0.25)(4.0) = 1.0 kg·m/s. Final P₁_x = (0.25)(3.0 cos 30°) = 0.75 × 0.866 = 0.65 kg·m/s.
• Step 2 (Conserve x-momentum): P_initial_x = P₁_x + P₂_x => 1.0 = 0.65 + P₂_x => P₂_x = 0.35 kg·m/s.
• Step 3 (Resolve y-component momentum): Initial P_y = 0. Final P₁_y = (0.25)(3.0 sin 30°) = 0.75 × 0.50 = +0.375 kg·m/s.
• Step 4 (Conserve y-momentum): P_initial_y = P₁_y + P₂_y => 0 = +0.375 + P₂_y => P₂_y = -0.375 kg·m/s (South).
• Step 5 (Find velocity components of puck 2): v₂_x = P₂_x / m₂ = 0.35 / 0.25 = 1.40 m/s. v₂_y = P₂_y / m₂ = -0.375 / 0.25 = -1.50 m/s.
• Step 6 (Calculate magnitude): v₂ = √((1.40)² + (-1.50)²) = √(1.96 + 2.25) = √4.21 ≈ 2.05 m/s.
• Step 7 (Calculate angle): θ = tan⁻¹(|-1.50| / 1.40) = tan⁻¹(1.071) ≈ 47.0° south of east.

Teacher's Strategy: Break 2D momentum into independent x and y tables before writing equations. This prevents sign confusion and ensures full marks for systematic presentation.

The 5 Most Common Mistakes Ontario Students Make in SPH4U

• Dropping direction signs in 1D problems: Forgetting that opposite directions require opposite signs.
• Assuming kinetic energy is always conserved: Believing inelastic collisions conserve Ek.
• Mixing x and y components: Adding horizontal and vertical momentum components directly without vector resolution.
• Confusing momentum (p = mv) with impulse (J = FΔt): Failing to recognize impulse represents change in momentum.
• Leaving vector magnitudes without directional angles on final answer statements.

University Admissions & Prerequisite Cutoff Impact

SPH4U Physics is required for engineering programs at McMaster, Waterloo, and U of T. Dynamics and momentum form the foundation of first-year university mechanics (PHYS 115 / PHY 180).

Diagnostic Readiness Check: Test Your Knowledge

• What is the SI unit of momentum?: Answer: kg·m/s or N·s.
• In a completely inelastic collision, what happens to the colliding objects?: Answer: They stick together and move with a shared common final velocity.
• If a net force acts on an object for time Δt, what quantity is transferred?: Answer: Impulse (which equals the change in momentum Δp).