A ball is thrown vertically up (taken as + Z-axis) from the ground. The correct momentum-height (p-h

Q: Why is the graph a parabola and not a straight line?

Because the relationship between velocity and height is quadratic ($v^2 \propto h$). Since momentum is directly proportional to velocity ($p = mv$), the relationship between momentum and height is also quadratic ($p^2 \propto h$), which forms a parabola.

Q: Why does the parabola open to the left?

If you express height $h$ as a function of momentum $p$, you get $h = A - Bp^2$. The negative coefficient in front of $p^2$ means the parabola opens towards the negative $h$-axis (to the left).

Q: What do the arrows on the graph represent?

The arrows represent the flow of time. As time passes, the ball first goes up (height increases, so the arrow points right) and then comes down (height decreases, so the arrow points left).

Q: Why is momentum negative in the bottom half of the graph?

Momentum is a vector quantity ($p = mv$). When the ball is falling down, its velocity is directed downwards. If we take the upward direction as positive, the downward velocity is negative, making the momentum negative.

Q: What does the vertex of the parabola represent physically?

The vertex of the parabola is the point where $p = 0$ and $h$ is at its maximum. This corresponds to the highest point in the ball's trajectory, where it momentarily stops before falling back down.

Question

A ball is thrown vertically up (taken as + Z-axis) from the ground. The correct momentum-height (p-h) diagram is

Options:

(A) (a)

(B) (b)

(C) (c)

(D) (d)

Sigma Percentile · Accepted Answer

The problem asks us to find the correct momentum-height ($p-h$) graph for a ball thrown vertically upwards. This is a beautiful exercise in translating physical motion into a mathematical phase space!

### Analyzing the Setup

Imagine standing on the ground and throwing a ball straight up into the air. We want to map out exactly how its momentum changes with its height.

To connect momentum and height, we first need to connect velocity and height. Remember our trusty third equation of motion?

$$v^2 = u^2 - 2gh$$

### The Master Equation

Now, momentum $p$ is simply mass times velocity ($p = mv$). So, let's multiply our entire kinematic equation by the mass squared ($m^2$) to bring momentum into the picture.

$$m^2v^2 = m^2u^2 - 2gm^2h$$

Substituting $p$ for $mv$, and $p_0$ for the initial momentum $mu$, we get a beautiful relationship:

$$p^2 = p_0^2 - 2gm^2h$$

If we rearrange this to solve for height $h$, we get:

$$h = \frac{p_0^2}{2gm^2} - \frac{1}{2gm^2} p^2$$

This is the exact mathematical equation of a parabola. Specifically, because of the negative sign in front of the $p^2$ term, it is a parabola that opens towards the left, or the negative $h$-axis.

### Tracing the Ascent

Let's trace the journey. As the ball flies upward, its height $h$ increases, but it slows down due to gravity.

So, its momentum is positive but shrinking to zero. On our graph, this traces the top half of the parabola. The arrow on this branch must point towards the right, indicating an increase in height.

### Tracing the Descent

At the very top, the ball stops for a split second ($v=0, p=0$) and then falls back down.

During the descent, the height $h$ decreases, and the ball speeds up in the downward (negative) direction. This means the momentum $p$ becomes increasingly negative.

This traces the bottom half of the parabola, moving back towards the vertical axis. The arrow on this branch must point towards the left, indicating a decrease in height.

### Final Conclusion

Looking at our options, we need a left-opening parabola where the top arrow points right (ascent), and the bottom arrow points left (descent).

**Option (d)** perfectly captures this entire physical story!

Animated Solution for Physics - Kinematics: A ball is thrown vertically up (taken as + Z-axis) from the ground. The correct momentum-height (p-h) diagram is

Select Answer:

Visualized Solution

$Physical Setup$

$Kinematic Equation$

$Introducing Momentum$

$Momentum-Height Relation$

$Equation of a Parabola$

$Ascent Phase$

$Descent Phase$

$Final Conclusion$

$Food for Thought$

The Sigma Insight: Motion Graphs

Analyzing the Setup

The Master Equation

Tracing the Ascent

Tracing the Descent

Final Conclusion

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