Q.A body of unit mass ($1\ \text{kg}$) moves in a plane, described by two velocity–time graphs. The $x$-velocity graph is triangular: $v_x$ rises linearly from $0$ at $t=0$ to $2\ \text{m s}^{-1}$ at $t=1\ \text{s}$, then falls linearly back to $0$ at $t=2\ \text{s}$, and remains $0$ for $t>2\ \text{s}$. The $y$-velocity graph rises linearly from $0$ at $t=0$ to $1\ \text{m s}^{-1}$ at $t=1\ \text{s}$, and then stays constant at $1\ \text{m s}^{-1}$ for $t>1\ \text{s}$. Find the force acting on the body as a function of time.

Newton's Second Law: The Law That Connects Force and Motion

Imagine you're pushing a shopping cart. If you push gently, it moves slowly. Push harder, and it speeds up faster. Now imagine the cart is full of groceries — even with the same push, it accelerates much more slowly than an empty cart. This everyday experience is exactly what Newton's Second Law captures.

The Intuition First

Two things matter when you push something:

1. How hard you push — the force you apply.
2. How heavy the object is — its mass.

The harder you push, the more the object speeds up. The heavier the object, the less it speeds up for the same push. So acceleration depends on both force and mass — and in opposite ways.

The Precise Statement

Newton's Second Law says:

The acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to its mass. The acceleration is in the same direction as the net force.

In one equation:

Or more commonly:

Where:

• $\vec{F}_{\text{net}}$ is the net force (the vector sum of all forces acting on the object) — measured in newtons (N)
• $m$ is the mass of the object — measured in kilograms (kg)
• $\vec{a}$ is the acceleration — measured in metres per second squared ($\text{m/s}^2$)

What This Really Means

Force causes acceleration, not velocity. A constant net force produces constant acceleration — meaning the velocity keeps changing at a steady rate. If you stop pushing, the net force becomes zero, and acceleration becomes zero (the object continues at constant velocity — that's Newton's First Law).

Mass is a measure of inertia. The more mass an object has, the harder it is to change its motion. A truck needs a much larger force than a bicycle to achieve the same acceleration.

Direction matters. Force and acceleration are vectors — they point the same way. If you push north, the acceleration is north. If multiple forces act, you must add them as vectors to find the net force.

A Simple Example

A 2 kg block is pushed with a net force of 10 N to the right.

The block accelerates at $5\ \text{m/s}^2$ to the right. Every second, its velocity increases by 5 m/s in that direction. …