91Ó°ÊÓ

When a projectile is launched at an angle to the ground with a certain beginning velocity, it follows a curved parabolic path. The horizontal range is the distance between the point of projection and the place of landing.

• Angle of the inclination of the ramp \(\theta = 32^\circ \).
• Speed of the motorcycle \(u = 40.0\,{\rm{m}}/{\rm{s}}\).
• Length of the bus \(L = 20.0\,{\rm{m}}\)

The horizontal range covered by the daredevil is,

\(R = \frac{{{u^2}\sin \left( {2\theta } \right)}}{g}\)

Here\(u\) is the initial velocity,\(\theta \) the angle of projection or angle of the ramp, and\(g\) is the acceleration due to gravity.

Substitute\(40{\rm{ m}}/{\rm{s}}\) for\(u\),\(32^\circ \) for\(\theta \), and\(9.8{\rm{ m}}/{{\rm{s}}^2}\) for\(g\),

\[\begin{array}{c}R = \frac{{{{\left( {40{\rm{ m}}/{\rm{s}}} \right)}^2} \times \sin \left( {2 \times 32^\circ } \right)}}{{\left( {9.8{\rm{ m}}/{{\rm{s}}^2}} \right)}}\\ = 146.74{\rm{ m}}\end{array}\]

The number of buses safely cleared by the daredevil is,

\(n = \frac{R}{L}\)

Here\(R\) is the horizontal range and\(L\) is the length of one bus.

Substitute\(146.74{\rm{ m}}\) for\(R\), and\(20.0{\rm{ m}}\) for\(L\),

\(\begin{array}{c}n = \frac{{146.74{\rm{ m}}}}{{20.0{\rm{ m}}}}\\ = 7.34\end{array}\)

Hence, the daredevil will safely clear the \(7\) buses.

The total length of the\(7\) buses is,

\({R_L} = 7 \times L\)

Here\(L\) is the length of one bus.

Substitute\(20.0{\rm{ m}}\) for\(L\),

\(\begin{array}{c}{R_L} = 7 \times \left( {20{\rm{ m}}} \right)\\ = 140{\rm{ m}}\end{array}\)

The margin of error is,

\(\gamma = \frac{{R - {R_L}}}{R} \times 100\% \)

Here\(R\) is the horizontal range and\({R_L}\) is the total length of\(7\) buses.

Substitute\(146.74{\rm{ m}}\) for\(R\), and\(140{\rm{ m}}\) for\({R_L}\),

\(\begin{array}{c}\gamma = \frac{{\left( {146.74{\rm{ m}}} \right) - \left( {140{\rm{ m}}} \right)}}{{\left( {146.74{\rm{ m}}} \right)}} \times 100\% \\ = 4.6\% \end{array}\)

Hence, the margin of error is \(4.6\% \). This is not enough, especially considering that air resistance was neglected by our calculation.