Free Fall, Relative Velocity and Calculus: Quick Revision

A brief note on Free Fall, Relative Velocity and Calculus. Try each question at the end before you open its answer.

Free fall

Without air, every body near the Earth has the same downward acceleration, $g = 9.8\ \text{m/s}^2$, whatever its mass. Taking up as positive:

• At the top of a vertical throw: $v = 0$, but $a = g$ downwards.
• Up and down take equal times; a ball comes back at the speed it was thrown.
• Thrown up at $u$: greatest height $\dfrac{u^2}{2g}$, time in the air $\dfrac{2u}{g}$.

Relative velocity

• Same direction: the difference of the speeds. Towards each other: the sum.
• Overtaking or passing: the distance to gain is both lengths.
• Two bodies falling together have no relative acceleration: their gap changes steadily.

Changing acceleration

The equations of motion are the special case of a constant $a$.

Worked examples

Thrown down. A stone is thrown down at $5\ \text{m/s}$ from a bridge $28\ \text{m}$ above a river. Down positive: $v^2 = (5\ \text{m/s})^2 + 2 \times 9.8\ \text{m/s}^2 \times 28\ \text{m} = 573.8\ \text{m}^2/\text{s}^2$, so $v = 24.0\ \text{m/s}$, and $t = \dfrac{24.0\ \text{m/s} - 5\ \text{m/s}}{9.8\ \text{m/s}^2} = 1.93\ \text{s}$.

From the position. For $x = t^3 - 4.5t^2 + 6t$ (SI units), $v = 3t^2 - 9t + 6 = 3(t - 1)(t - 2)$: the body stops at $1\ \text{s}$ and $2\ \text{s}$, and between them it moves back.

Practice with new numbers

A program to try

Two stones leave the same bridge at once: one dropped, one thrown down.

Mistakes to avoid

• "The acceleration is zero at the top." It is $g$, downwards, all the way.
• Heavier falls faster. Not without air.
• The order in relative velocity. $v_{AB} = v_A - v_B$; swapping turns the answer round.
• Using $v = u + at$ when $a$ changes. Integrate instead.
• Forgetting the starting velocity when integrating the acceleration.

Check yourself

1. A ball is thrown up at $12.25\ \text{m/s}$. How high does it rise?

2. A stone is dropped from rest. How far does it fall in the first $0.5\ \text{s}$?

3. Two buses move the same way at $16\ \text{m/s}$ and $11\ \text{m/s}$. How fast does the first move, as seen from the second?

4. For $v = 6t - t^2$ (SI units), when is the acceleration zero, and what is the velocity then?

5. A body starts from rest with $a = 2t$ (SI units). What is its velocity after $5\ \text{s}$?