Question

a car steadily speeds up from 6m/s to 30m/s. determine all unknowns. v0 = 6 m/s v = m/s vf = 30 m/s

Explanation:

1. First, assume the motion is a uniformly - accelerated linear motion. The basic equations of uniformly - accelerated linear motion are:

• \(v = v_0+at\), \(x = v_0t+\frac{1}{2}at^{2}\), \(v^{2}-v_0^{2}=2ax\). Here, \(v_0 = 6m/s\) is the initial velocity and \(v_f=v = 30m/s\) is the final velocity.
• Acceleration \(a\):
• From \(v = v_0+at\), if we assume the time \(t\) is known, \(a=\frac{v - v_0}{t}\). But if \(t\) is not given, from \(v^{2}-v_0^{2}=2ax\), we can also express \(a=\frac{v^{2}-v_0^{2}}{2x}\). Without information about time \(t\) or displacement \(x\), we can't calculate a numerical value for \(a\). However, in terms of variables, \(a=\frac{v - v_0}{t}\) (where \(v = 30m/s\), \(v_0 = 6m/s\)).
• Time \(t\):
• From \(v = v_0+at\), we can solve for \(t\): \(t=\frac{v - v_0}{a}\). Since \(v = 30m/s\) and \(v_0 = 6m/s\), \(t=\frac{30 - 6}{a}=\frac{24}{a}\) (when \(a

eq0\)).

• Displacement \(x\):
• From \(v^{2}-v_0^{2}=2ax\), we can solve for \(x\): \(x=\frac{v^{2}-v_0^{2}}{2a}\). Substituting \(v = 30m/s\) and \(v_0 = 6m/s\), we get \(x=\frac{30^{2}-6^{2}}{2a}=\frac{900 - 36}{2a}=\frac{864}{2a}=\frac{432}{a}\) (when \(a

eq0\)).

1. If we assume the motion is uniformly - accelerated and no other information is given:

• The unknowns are acceleration \(a\), time \(t\), and displacement \(x\). The relationships for these unknowns are:
• Acceleration \(a=\frac{v - v_0}{t}=\frac{30 - 6}{t}=\frac{24}{t}\) (using \(v = v_0+at\)) or \(a=\frac{v^{2}-v_0^{2}}{2x}=\frac{30^{2}-6^{2}}{2x}=\frac{864}{2x}=\frac{432}{x}\) (using \(v^{2}-v_0^{2}=2ax\)).
• Time \(t=\frac{v - v_0}{a}=\frac{24}{a}\).
• Displacement \(x=\frac{v^{2}-v_0^{2}}{2a}=\frac{432}{a}\).

Answer:

The unknowns are acceleration \(a\), time \(t\), and displacement \(x\) with relationships \(a=\frac{24}{t}=\frac{432}{x}\), \(t = \frac{24}{a}\), \(x=\frac{432}{a}\) (assuming uniformly - accelerated linear motion and no other information given).