The image shows a physics problem and its solution about 2D motion kinematics.
Skill Builder Problem:
Given: x = 3t - 2t³ and y = 15t (metres, seconds)
Find acceleration at t = 5s.
Solution:
x-direction:
- vₓ = dx/dt = 3 - 6t²
- aₓ = dvₓ/dt = -12t
At t = 5s: aₓ = -12(5) = -60 m/s²
y-direction:
- vᵧ = dy/dt = 15 m/s (constant)
- aᵧ = dvᵧ/dt = 0
Net acceleration:
a_net = aₓ î + aᵧ ĵ = -60î m/s²
The answer is (b) -60î m/s²
Example 6 (partially visible):
Velocity changes from (3î + 4ĵ) m/s to (6î + 5ĵ) m/s in 2 s. Find magnitude and direction of average acceleration.
Change in velocity: Δv = (6-3)î + (5-4)ĵ = 3î + 1ĵ m/s
Average acceleration: a = Δv/t = (3î + ĵ)/2 = 1.5î + 0.5ĵ m/s²
- Magnitude: |a| = √(1.5² + 0.5²) = √(2.25 + 0.25) = √2.5 ≈ 1.58 m/s²
- Direction: θ = tan⁻¹(0.5/1.5) = tan⁻¹(1/3) ≈ 18.4° above the x-axis
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