Submitted by Wenfei on Thu, 03/22/2018 - 09:17
Forums:
The question says the particle is projected in a direction that makes an angle 60 with OB, but it is actually the "shadow" in the horizontal field that makes the angle 60 with OB according to the solution. I think it should be made clear in the question to avoid confusion.
And for the last part of the
And for the last part of the question, the solution says the particle has travelled just 5/64 metres vertically, but shouldn't it be 5/8*sqrt(5)-5/64?
Vertical motion
Let $t_P=\frac{\sqrt{5}} 2$ (i.e. $t_P$ is the time when the particle is at $P$. Then we are interested in the vertical displacement of the particle at $t=t_P+\frac 1 8$ relative to $P$. Using the fact that the vertical displacement is given by $(5 \sqrt{5} t - 5t^2)$ then the relative displacement is:
\[t_P+\frac 1 8
\Big( 5 \sqrt{5} (t_P+\tfrac 1 8) - 5(t_P+\tfrac 1 8)^2 \Big)-\Big(5 \sqrt{5} t_P - 5t_P^2 \Big)\\
=5\sqrt{5} \times \tfrac 1 8 - 5 \Big(t_P^2+\tfrac 1 4 t_P+\tfrac 1 {64} \Big) + 5t_P^2\\
=5\sqrt{5} \times \tfrac 1 8 - \tfrac 5 4 \times t_P -\tfrac 5 {64}\\
=5\sqrt{5} \times \tfrac 1 8 - \tfrac 5 4 \times \tfrac {\sqrt{5}}2 -\tfrac 5 {64}\\
=-\tfrac 5 {64}
\]
Vertical motion
I understand. Thank you.
Mechanics Q 3
I don't think we should change the wording of the STEP question, as this was a real question used in a paper. Reading through it, they have been careful to describe the angle the particle makes with the horizontal first, which was done to try and avoid confusion.
Trying to describe motion in 3D in Cartesian coordinates is tricky, and the wording of this question would have been very carefully thought about. It might have been the case that the idea of a "shadow" was mentioned, but this may have been thought to be even more confusing (and possibly not well defined - the position of the shadow would depend on where the sun is!)
I will edit the solutions to try and make the situation clearer.
Mechanics Q 3
Thank you for your reply. The confusion is that according to the wording of the question the angle between the direction of the velocity of the particle and OB is 60 but it is actually not (should be greater than 60). I understand this was a real question so we are not going to change it.