Step-wise force and v–t graph

Problem

A block moves on a horizontal plane and the magnitude of the applied horizontal force varies over time as shown in the figure. Sketch qualitatively the velocity–time graph, assuming the block starts from rest and the plane is smooth.

Solution

Step 1 — From force to acceleration. The plane is smooth, so the only horizontal force is the applied one and the second law gives $a = F/m$. Since the mass $m$ is constant, the acceleration is proportional to the force: where $F$ is constant, so is $a$, and the velocity grows linearly with time (uniformly accelerated motion). We distinguish three intervals:

• $0 \le t \le t_1$: constant, small force $F_1$ → constant acceleration $a_1 = F_1/m$ → $v$ grows with a shallow slope;
• $t_1 \le t \le t_2$: constant, larger force $F_2 > F_1$ → larger acceleration $a_2 = F_2/m$ → $v$ grows with a steeper slope;
• $t \ge t_2$: zero force → zero acceleration → $v$ remains constant (horizontal plateau), equal to the value reached.

Step 2 — Constructing the $v(t)$ graph. The velocity is the integral (the area under the $a$–$t$ graph): it starts from zero, rises as a line of moderate slope over the first stretch, then as a steeper line over the second stretch (joining smoothly, without jumps — the velocity cannot change abruptly), and finally continues horizontally once the force vanishes. It never returns to zero: on the smooth plane, once the force is removed, the block keeps by inertia the velocity it has acquired.

$\ev{v(t) \text{ increasing and piecewise: two increasingly steep straight stretches, then a horizontal plateau.}}$

Links

Topics: Dynamics Concepts: Newton’s second law Skills: Reading graphs