Ncert Exercises & Solutions

Figure shows $(v_{x}, t)$ $(v_{x}, t)$ and $(v_{y}, t)$ $(v_{y}, t)$ diagrams for a body of unit mass. Find the force as a function of time.

Hint: Use acceleration, a(t) =

\frac{dv (t)}{dt} .

$\frac{dv (t)}{dt} .$

Step 1: Find force on the body aong x-axis.

Consider figure (a)

\begin{array}{l} v_{x} = 2 t for 0 < t < 1 s \\ = 2 (2 - t) for 1 < t < 2 s \\ = 0 for t > 2 s \end{array}

$\begin{array}{l} v_{x} = 2 t for 0 < t < 1 s \\ = 2 (2 - t) for 1 < t < 2 s \\ = 0 for t > 2 s \end{array}$

F_{x} = {ma}_{x} = m \frac{{dv}_{x}}{dt}

$F_{x} = {ma}_{x} = m \frac{{dv}_{x}}{dt}$

\begin{array}{l} = 1 \times 2 for 0 < t < 1 s \\ = 1 (- 2) for t > 2 s \end{array}

$\begin{array}{l} = 1 \times 2 for 0 < t < 1 s \\ = 1 (- 2) for t > 2 s \end{array}$

Step 2: Find the force on the body along the y-axis.

From figure (b)

v_{y} = t for 0 < t < 1 s

$v_{y} = t for 0 < t < 1 s$

= 1 for t > 1 s

$= 1 for t > 1 s$

\begin{array}{l} F_{y} = {ma}_{y} = m \frac{{dv}_{y}}{dt} \\ = 1 \times 1 for 0 < t < 1 s \\ = 0 for t > 1 s \\ Step 3 : Find net force on the body . \\ Hence, \vec{F} = F_{x} \hat{i} + F_{y} \hat{j} \\ = 2 \hat{i} + \hat{j} for 0 < t < 1 s \\ = - 2 \hat{i} for 1 < t < 2 s \\ = 0 for t > 2 s \end{array}

$\begin{array}{l} F_{y} = {ma}_{y} = m \frac{{dv}_{y}}{dt} \\ = 1 \times 1 for 0 < t < 1 s \\ = 0 for t > 1 s \\ Step 3 : Find net force on the body . \\ Hence, \vec{F} = F_{x} \hat{i} + F_{y} \hat{j} \\ = 2 \hat{i} + \hat{j} for 0 < t < 1 s \\ = - 2 \hat{i} for 1 < t < 2 s \\ = 0 for t > 2 s \end{array}$

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