Set 3 Problem number 8

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

An object, initially at rest, is acted upon by a net force of 56.7 Newtons. The object has mass 8.1 kilograms.

• What will be its acceleration, and what will be its speed after the first 4.5 seconds of acceleration?
• How far will it travel up to this time?

Solution

The acceleration of an object of mass 8.1 Kg under the influence of a 56.7 Newton force will be

• a=F/m=( 56.7 Newtons)/( 8.1 Kg)= 7 meters per second per second.

In 4.5 seconds the velocity change will be

• `dv = 4.5( 7 m/s) = 31.5 meters per second.

Since the object starts from rest this will also be the final velocity.

The average velocity will be the average of this final velocity and zero, or ( 31.5 + 0)/2 meters per second = 15.75 meters per second.

At this average velocity, in 4.5 seconds the object will move 70.875 meters.

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 0 + `dv = (F / m) `dt.
• The average velocity will be the average of its initial and final velocities, or (F / (2m) ) * `dt.
• The distance moved will therefore be

• `ds = vAve * `dt = (F / (2m) ) * `dt².

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).
• From initial and final velocities we obtain average velocity (the 'red' triangle) and from average velocity and `dt we obtain the displacement `ds.

Figure(s)

Figure(s)