Question pattern: Multi-stage motion
Original question 1 of 1 in this pattern
Multi-stage motion
A cyclist passes point P at 5.0 m s⁻¹. The cyclist then accelerates uniformly at 1.2 m s⁻² for 6.0 s, travels at the resulting constant speed for 4.0 s, and finally decelerates uniformly at 2.0 m s⁻² until coming to rest.
Calculate:
a) the maximum speed;
b) the total distance travelled from P until the cyclist stops;
c) the average speed for the complete journey.
Worked answer
Solution
- Step 1
Maximum speed:
v = 5.0 + 1.2 × 6.0 = 12.2 m s⁻¹. - Step 2
First-stage distance:
s₁ = 5.0(6.0) + 1/2(1.2)(6.0²) = 51.6 m. - Step 3
Constant-speed distance:
s₂ = 12.2 × 4.0 = 48.8 m. - Step 4
Deceleration time:
t₃ = 12.2 / 2.0 = 6.10 s. - Step 5
Final-stage distance:
s₃ = ((12.2 + 0) / 2) × 6.10 = 37.2 m. - Step 6
Total distance:
137.6 m. - Step 7
Total time:
16.1 s. - Step 8
Average speed:
137.6 / 16.1 = 8.55 m s⁻¹.
Final answer
Maximum speed = 12.2 m s⁻¹; total distance ≈ 138 m; average speed = 8.55 m s⁻¹.
Common mistake
Using the mean of the initial and final velocities as the average speed for the entire multi-stage journey.
Concepts tested
- multi-stage motion
- acceleration
- velocity-time reasoning
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