SPH4U Grade 12 Physics: 2D Kinematics & Dynamics Practice Guide [Worked Solutions]

In Ontario's SPH4U Grade 12 University Physics, 2D Kinematics and Dynamics model the motion of objects under gravitational and contact forces. Horizontal motion proceeds at constant velocity (ax = 0), while vertical motion experiences constant gravitational acceleration (ay = -9.8 m/s²). Dynamics resolves free-body diagram vectors on inclined planes using Newton's Second Law ΣF = ma.

Official Ontario Curriculum Breakdown: SPH4U

• Ministry Course Code: SPH4U (Physics, Grade 12, University Preparation)
• Curriculum Strands: Dynamics, Energy and Momentum, Gravitational, Electric and Magnetic Fields, The Wave Nature of Light, Revolutions in Modern Physics
• Prerequisite Course: SPH3U (Grade 11 Physics, University Preparation)
• Recommended Math: MCR3U completed; concurrent enrollment in MHF4U or MCV4U recommended
• Post-Secondary Programs: Mandatory for Engineering, Physics, Astrophysics, and Physical Sciences

Worked Example 1: 2D Angled Projectile Launched from a Cliff

Problem: A projectile is launched from the top of a 30.0 m high cliff with an initial velocity of 25.0 m/s at an angle of 35.0° above the horizontal. Calculate the total flight time and the horizontal distance (range) where it strikes the ground below. (Use g = 9.80 m/s²).

• Step 1 (Resolve velocity into components): v₀ₓ = 25.0 · cos(35.0°) = 20.48 m/s. v₀ᵧ = 25.0 · sin(35.0°) = 14.34 m/s.
• Step 2 (Set up vertical kinematic equation): Choose upward as positive. y₀ = +30.0 m, y = 0 m (ground), aᵧ = -9.80 m/s², v₀ᵧ = +14.34 m/s. Equation: y = y₀ + v₀ᵧ·t + 1/2·aᵧ·t² ⇒ 0 = 30.0 + 14.34t - 4.90t².
• Step 3 (Solve the quadratic equation for t): Rearrange: 4.90t² - 14.34t - 30.0 = 0. Using quadratic formula: t = [14.34 ± √(14.34² - 4(4.90)(-30.0))] / [2(4.90)] = [14.34 ± √(205.64 + 588.0)] / 9.80 = [14.34 ± √793.64] / 9.80 = [14.34 ± 28.17] / 9.80.
• Step 4 (Select positive physical time): t = (14.34 + 28.17) / 9.80 = 42.51 / 9.80 = 4.34 seconds.
• Step 5 (Calculate horizontal range): Since aₓ = 0: Δx = v₀ₓ · t = (20.48 m/s) · (4.34 s) = 88.9 meters.
• Conclusion: Total flight time is 4.34 seconds, and the projectile travels 88.9 meters horizontally.

Teacher's Strategy: Never mix horizontal and vertical variables. Time t is the unique shared scalar parameter connecting independent horizontal and vertical equations.

Worked Example 2: Dynamics of a Block on an Incline with Friction

Problem: A 5.0 kg block accelerates down a 30° ramp. The coefficient of kinetic friction between the block and ramp is μₖ = 0.20. Calculate the magnitude of the block's acceleration down the ramp.

• Step 1 (Resolve gravity parallel and perpendicular to ramp): Choose parallel down ramp as +x; perpendicular to ramp as +y. Fgₓ = mg · sin(30°) = (5.0)(9.80)(0.50) = 24.5 N. Fgᵧ = mg · cos(30°) = (5.0)(9.80)(0.866) = 42.4 N.
• Step 2 (Determine Normal Force): Since there is no acceleration in the y-direction: ΣFᵧ = 0 ⇒ F_N - Fgᵧ = 0 ⇒ F_N = 42.4 N.
• Step 3 (Calculate Kinetic Friction): Friction opposes motion up the ramp: fₖ = μₖ · F_N = (0.20)(42.4 N) = 8.48 N.
• Step 4 (Apply Newton's Second Law along the ramp): ΣFₓ = m · a ⇒ Fgₓ - fₖ = m · a ⇒ 24.5 N - 8.48 N = (5.0 kg) · a.
• Step 5 (Solve for acceleration): 16.02 N = 5.0 · a ⇒ a = 3.20 m/s² down the ramp.

Teacher's Strategy: Always draw a Free-Body Diagram (FBD) with tilted axes. Students lose marks when they assume F_N = mg instead of F_N = mg cos(θ).

The 5 Most Common Mistakes Ontario Students Make in SPH4U

• Mixing horizontal and vertical components (e.g. using vertical acceleration g in horizontal displacement equations).
• Assuming normal force equals mg on an inclined plane (it is mg cos θ when no other vertical forces act).
• Forgetting that static friction is an inequality (f_s ≤ μ_s F_N) while kinetic friction is an exact value (f_k = μ_k F_N).
• Sign errors in projectile kinematics when launch is upward (+v₀ᵧ) and displacement is downward (-Δy).
• Rounding intermediate numbers too early in multi-step physics calculations.

University Admissions & Prerequisite Cutoff Impact

SPH4U Physics is one of the highest-weight courses in the Waterloo and U of T Engineering Admission Information Form (AIF) evaluations. Mastering kinematics, dynamics, and energy conservation in Grade 12 directly correlates with success in first-year engineering mechanics.

Diagnostic Readiness Check: Test Your Knowledge

• What is the vertical velocity of a projectile at the peak of its trajectory? — Answer: 0 m/s (horizontal velocity v_x remains constant).
• What does the area under a Force vs. Position (F-d) graph represent? — Answer: Work done (in Joules).
• If net force on an object is zero, does it mean it is stationary? — Answer: No, it moves with constant velocity in a straight line (Newton's First Law).