Which of the following equations of motion would be used to calculate the displacement given the initial and final velocities as well as the acceleration?

Question

BlackTom AI · Accepted Answer

The SUVAT equation linking v, u, a and s is v^2 = u^2 + 2as.

Relevant knowledge points:
- SUVAT (constant acceleration) equations relate displacement s, initial velocity u, final velocity v, acceleration a, and time t.
- To calculate displacement when u, v and a are known, we need an equation that does not involve time t.

Step-by-step reasoning and derivation:
1. Start with two standard SUVAT equations:
   v = u + at
   s = ut + (1/2) a t^2
2. Eliminate t by solving the first for t:
   t = (v − u)/a
3. Substitute this t into the displacement equation:
   s = u * [(v − u)/a] + (1/2) a * [(v − u)/a]^2
4. Simplify term by term:
   s = [u(v − u)]/a + (1/2) a * (v − u)^2 / a^2
     = [u(v − u)]/a + (1/2) (v − u)^2 / a
5. Combine over common denominator a:
   s = { u(v − u) + (1/2)(v − u)^2 } / a
6. Expand numerator:
   u v − u^2 + (1/2)(v^2 − 2uv + u^2)
   = (1/2) v^2 − (1/2) u^2
7. Thus s = [ (1/2)(v^2 − u^2) ] / a = (v^2 − u^2) / (2a)
8. Rearranging gives v^2 = u^2 + 2as.

Why the correct answer is correct:
- v^2 = u^2 + 2as directly relates initial and final velocities, acceleration, and displacement without involving time, so it is the appropriate formula when u, v and a are given and s is required.

Why the other options are incorrect:
- b. v = u + at gives v in terms of u, a and t but does not give displacement s.
- c. s = ½(u + v)t expresses s in terms of u, v and t; it still requires time t.
- d. s = ut + ½at^2 gives s but depends on t, which is not provided.

Conclusion:
Therefore the correct choice is a. v^2 = u^2 + 2as, which matches the provided answer.

Which of the following equations of motion would be used to calculate the displacement given the initial and final velocities as well as the acceleration?单项选择题

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