Velocity (m/s)-time (s) graph is plotted for the motion of two cars A and B for 10 s, when the cars are initially at rest and acquire final velocities as VA and VB, respectively. The respective straight lines of the graph make angles 30° and 60°, respectively with the x-axis (time). The difference in velocities VB and VA is
\(20/\sqrt{3}\) m/s
On a velocity-time graph, the slope of the straight line gives the acceleration, i.e. \(a = \tan\theta\), where θ is the angle the line makes with the time axis. Since both cars start from rest, the final velocity after time t is \(v = a \times t = t\tan\theta\).
For car A: \(\theta_A = 30^\circ\), so \(V_A = 10 \times \tan30^\circ = 10 \times \dfrac{1}{\sqrt{3}} = \dfrac{10}{\sqrt{3}}\ \text{m/s}\).
For car B: \(\theta_B = 60^\circ\), so \(V_B = 10 \times \tan60^\circ = 10\sqrt{3}\ \text{m/s}\).
The difference is \(V_B - V_A = 10\sqrt{3} - \dfrac{10}{\sqrt{3}} = \dfrac{10 \times 3 - 10}{\sqrt{3}} = \dfrac{20}{\sqrt{3}}\ \text{m/s}\).
Hence, the difference in velocities is \(20/\sqrt{3}\) m/s.
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