Mechanics of a falling body

A short derivation of vertical motion, from Newton’s second law to the conservation of energy.

A falling body is one of the simplest systems we can study. A few carefully chosen assumptions can take us a surprisingly long way.

Framing the problem

Imagine a body released above a reference plane. Let $y(t)$ denote its height at time $t$, with the upward direction taken as positive.

We make three assumptions:

• The gravitational field is uniform.
• Air resistance is negligible.
• The body can be treated as a point mass.

Near Earth’s surface, these approximations work well over modest distances.

Equations of motion

Newton’s second law relates the net force to acceleration. Since gravity acts downward, it enters with a minus sign:

$$m\frac{d^2y}{dt^2} = -mg$$

The mass cancels. Integrating once gives velocity; integrating again gives position. The constants are the initial velocity $v_0$ and initial height $y_0$.

$$v(t) = v_0 - gt$$ $$y(t) = y_0 + v_0t - \frac{1}{2}gt^2$$

For a body released from rest at height $h$, the time of flight is $t_f = \sqrt{\frac{2h}{g}}$. It does not depend on the mass.

Energy form

In a conservative gravitational field, the sum of kinetic and potential energy stays constant:

$$\frac{1}{2}mv^2 + mgy = \frac{1}{2}mv_0^2 + mgy_0$$

This formulation is useful when we care about the speed at a given height, rather than the time taken to reach it.

If velocity doubles

The kinetic energy is $E_k = \frac{1}{2}mv^2$. Doubling velocity gives:

$$E_k' = \frac{1}{2}m(2v)^2 = 4E_k$$

Therefore the kinetic energy increases by a factor of four.

A worked example

Release a sphere from rest at a height of 20 m, taking $g = 9.81\,\mathrm{m/s^2}$. A few lines of Python give the flight time and impact speed.

from math import sqrt

height = 20       # metres
gravity = 9.81    # metres / second²

flight_time = sqrt(2 * height / gravity)
impact_speed = gravity * flight_time

print(f"Time: {flight_time:.2f} s")   # 2.02 s
print(f"Speed: {impact_speed:.1f} m/s")  # 19.8 m/s

Quantity	Result
Time of flight	2.02 s
Impact speed	19.8 m/s

The sphere reaches the reference plane in 2.02 seconds, moving at approximately 19.8 m/s. In an experiment, drag and measurement uncertainty will introduce small differences.

Closing note

The value of a derivation is not only its answer. It makes the assumptions visible, so each can be revisited when the problem calls for a richer model. Return to the equations of motion to review the starting point.