Q.A car starts from rest and moves on a surface with uniform acceleration. Draw the graph of kinetic energy versus displacement. What information can you get from that graph?

Kinetic Energy and Speed: The Intuition

Imagine pushing a heavy box across the floor. The harder you push, the faster it moves. But here's the surprising part: doubling the speed does not require double the work — it requires four times the work. That's the core of the kinetic energy–speed relation.

Why? Because kinetic energy isn't about how fast you're moving — it's about how much effort it took to get you moving that fast. And effort (work) depends on both force and distance. When you push something to a higher speed, you have to apply force over a longer distance, and that extra distance multiplies the work required.

The Precise Statement

The kinetic energy $K$ of an object of mass $m$ moving with speed $v$ is:

This is the kinetic energy–speed relation. The key point: kinetic energy is proportional to the square of the speed, not the speed itself.

Why the Square? A Simple Derivation

Start from Newton's second law: $F = ma$. If a constant force $F$ acts on an object initially at rest over a displacement $s$, the work done is $W = F s$.

From kinematics, for constant acceleration starting from rest: $v^2 = 2 a s$. So $s = \frac{v^2}{2a}$.

Substitute into work:

That work becomes the object's kinetic energy. The square comes from the kinematic relation $v^2 = 2as$ — a direct consequence of how distance and speed are linked under constant acceleration.

What This Means in Practice

This explains why:

• A car crash at 100 km/h is four times as destructive as one at 50 km/h (four times the energy to dissipate).
• Braking distance quadruples when speed doubles (because brakes must do four times the work).
• A bullet at twice the speed penetrates much deeper than twice as far.

Units

In SI units:

• Mass $m$ in kilograms (kg) …