Problem Statement
A car starts from rest, accelerates at $w = 5.0\, ext{m/s}^2$, then moves uniformly, then decelerates at the same rate $w$ to rest. Total time $ au = 25\, ext{s}$, mean velocity $\langle v angle = 72\, ext{km/h}$. How long does it move uniformly?
Given Information
- $w = 5.0\, ext$
- $ au = 25\, ext$
- $\langle v angle = 72\, ext$
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: Let $t_0$ = time of each acceleration/deceleration phase, $t_1$ = uniform phase time.
Step 2 — Apply the relevant physical law or equation: Constraint: $2t_0 + t_1 = au$
Step 3 — Solve algebraically for the unknown: Peak speed: $v_{\max} = wt_0$
Step 4 — Substitute numerical values with units: Total distance: $d = wt_0(t_0+t_1)$
Step 5 — Compute and check the result: Mean velocity: $\langle v
angle = wt_0(t_0+t_1)/ au$
Step 6: Converting: $\langle v
angle = 72\, ext{km/h} = 20\, ext{m/s}$. Substituting:
Worked Calculation
$$wt_0( au – t_0) = \langle v
angle au$$
$$5t_0^2 – 5(25)t_0 + 20(25) = 0$$
$$t_0^2 – 25t_0 + 100 = 0$$
Let $t_0$ = time of each acceleration/deceleration phase, $t_1$ = uniform phase time.
Constraint: $2t_0 + t_1 = au$
Peak speed: $v_{\max} = wt_0$
Total distance: $d = wt_0(t_0+t_1)$
Mean velocity: $\langle v
angle = wt_0(t_0+t_1)/ au$
Converting: $\langle v
angle = 72\, ext{km/h} = 20\, ext{m/s}$. Substituting:
$$wt_0( au – t_0) = \langle v
angle au$$
$$5t_0^2 – 5(25)t_0 + 20(25) = 0$$
$$t_0^2 – 25t_0 + 100 = 0$$
$$t_0 = rac{25 \pm 15}{2}$$
Take $t_0=5\, ext{s}$:
$$t_1 = 25 – 2(5) = oxed{15\, ext{s}}$$
Answer
$$\boxed{t_1 = 25 – 2(5) = oxed{15\, ext{s}}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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