- #1

brotherbobby

- 651

- 156

- Homework Statement
- A ball is thrown vertically up. Its distance from a fixed point varies with time according to the given graph (see below). Calculate the ##\text{velocity of projection}## of the ball.

- Relevant Equations
- For uniformly accelerated motion under gravity, taking ##g = +9.8\;\text{m/s}^2##, we have the position after a time ##t##, ##y(t) = y_0+v_0t-\frac{1}{2}gt^2\;\text{(I)}##, and the velocity ##v(t)=v_0-gt\; \text{(II)}##. The velocity can also be expressed as a funtion of the position ##y## from the origin, ##v^2(y)=v^2_0-2gy\;\text{(III)}##.

Additionally, if the given fixed point doesn't lie along the path of motion of the ball, then ##\vec s=\vec{r}-\vec {r'}##, where ##\vec s## is the position vector of the ball from the point and ##\vec{r},\vec{r'}## are those of the ball and the point from some origin (say from the point of projection).

**I copy and paste the statement of the problem to the right as it appeared on the website. Given below is the graph of the ball as its distance from a fixed point with time.
Statement of the problem : **

**Attempt :**Where does this fixed point, say ##\text{P}## lie?

Imagine the fixed point lied along the path of motion of the ball. It can't be below the point of projection ##\text{O}##, as distance of separation decreases initially. It can't be somewhere between ##\text{O}## and ##\text{H}##, the maximum height, since in that case the distance should go to 0 for some time ##t## where the ball passes through the point. It can lie above the maximum height ##\text{H}##. In that case, the distance of separation will reduce for some time and begin to increase again, as it does on the graph. But it should go up to the same value for some time ##t## later as for ##t=0## which it does not!

I suspect the fixed point of observation lies

__outside__the path of motion of the ball, as shown in the diagram below. Additionally, for reference, I label points along the graph.

I put points along the motion of the ball that correspond to those in the graph in yellow on black background.

**But this diagram cannot be correct.**

From the graph, at

**B**, where the distance from

**P**is minimum, the body is at rest (momentarily), meaning its distance of separation isn't changing with time. But from my motion diagram on the right, the distance of separation ##s## is changing with time. So they contradict. Likewise for

**D**.

If you put the observation point

**P**anywhere else, it would violate the distance of separation as given in the graph.

I couldn't do better than this.

A hint or suggestion would be very welcome.