Set 3 Problem number 7

An object, initially at rest, is acted upon by a net force of 86 Newtons. The object has mass 13 kilograms.

What will be its acceleration, and what will be its speed after the first 10 seconds of acceleration?

Solution

The first question is asking for the rate at which velocity increases, which is a=F/m=( 86 Newtons)/( 13 Kg)= 6.615385 meters per second per second.

Changing by 6.615385 meters per second every second, in 10 seconds the increase will be ( 10 sec)( 6.615385 m/sec/sec) = 66.15385 m/s.
Since the object started from rest, this will also be its velocity after 10 seconds.

Generalized Solution

The rate at which velocity changes, or the acceleration, is a = F / m.

During a time interval `dt, the velocity will therefore change by a `dv = (F / m) `dt.
If the object starts from rest its final velocity will therefore be

final velocity from rest = vf = 0 + `dv = (F / m) `dt.

Explanation in terms of Figure(s), Extension

The first figure below depicts a 'flow' triangle for an object of mass m subjected to a force F.

The relationship a = F / m can be understood as saying that greater force implies proportionally greater acceleration for a given mass while greater mass implies proportionally less acceleration for a given force.
The relationship F = m * a can be understood as saying that

to achieve a given acceleration a greater force must be exerted on a greater mass and that

for a given mass a greater acceleration will require a greater force.

The relationship m = F / a tells us that for a given observed acceleration the greater the force the greater the mass being accelerated, and for a given applied force a greater acceleration implies that less mass is being accelerated.

The second figure depicts a 'flow' diagram for an object of mass m, initially with velocity v0, subjected to a force F for time interval `dt.

From the force and the mass we deduce the acceleration (the 'blue' triangle).
From the acceleration and the time interval we deduce the change `dv in velocity (the 'green' triangle).
From the change in velocity and initial velocity we obtain the final velocity (the 'purple' triangle).