An object with an initial velocity, [math: u], of 35.8 ms-1 obtain a displacement, [math: x], of 37.3 m with an acceleration, [math: a], of 9.7 ms-2. Find the time, [math: t], taken for this motion.

Question

BlackTom AI · Accepted Answer

Use the kinematic equation x = ut + (1/2) a t^2.

1) Relevant knowledge points and concepts
- For motion with constant acceleration, displacement x, initial velocity u, acceleration a, and time t are related by x = ut + (1/2) a t^2.
- Solving for t leads to a quadratic equation; quadratic formula gives two roots. Physically we pick the nonnegative time.

2) Complete reasoning process step by step
- Start from x = ut + (1/2) a t^2 and rearrange to standard quadratic form in t.
- Apply the quadratic formula t = [-u ± sqrt(u^2 + 2ax)] / a (or equivalently use the full quadratic formula for coefficients).

3) Calculations with intermediate steps
Given u = 35.8 m/s, x = 37.3 m, a = 9.7 m/s^2.
Write 0 = (1/2)a t^2 + u t - x, so:
(1/2)a = 4.85, giving 4.85 t^2 + 35.8 t - 37.3 = 0.

Use quadratic formula t = [-35.8 ± sqrt((35.8)^2 - 4·4.85·(-37.3))] / (2·4.85).

Compute discriminant:
(35.8)^2 = 1281.64.
4·4.85·37.3 = 19.4·37.3 = 723.62.
Discriminant = 1281.64 + 723.62 = 2005.26.
sqrt(discriminant) ≈ 44.78.

Denominator 2·4.85 = 9.7.
Thus t = [-35.8 ± 44.78] / 9.7.

Two roots:
t1 = (-35.8 + 44.78)/9.7 = 8.98/9.7 ≈ 0.9268 s ≈ 0.93 s.
t2 = (-35.8 - 44.78)/9.7 = -80.58/9.7 ≈ -8.312 s (physically not applicable for forward time).

4) Why the correct answer is correct
The positive root 0.93 s satisfies the kinematic equation for the given u, a, and x. The negative root is a mathematically valid solution but represents a time before the chosen t = 0, so it is discarded for forward-time motion.

5) Why other options are incorrect
- a. 8.31 s corresponds to the magnitude of the negative root but is negative in sign, so not a valid forward time.
- b. 6.13 s does not satisfy the quadratic solution.
- c. -1.26 s is neither the computed negative root nor the positive physical root.

6) Conclusion
The correct time is 0.93 s, which matches the provided answer.

An object with an initial velocity, [math: u], of 35.8 ms-1 obtain a displacement, [math: x], of 37.3 m with an acceleration, [math: a], of 9.7 ms-2. Find the time, [math: t], taken for this motion.数值题

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