Write down the dimensions of velocity, acceleration and force.
A ball of mass \(m\) is thrown vertically upwards with initial velocity \(U\). When the velocity of the ball is \(v\), it experiences a force \(\lambda v ^ { 2 }\) due to air resistance where \(\lambda\) is a constant.
Find the dimensions of \(\lambda\).
A formula approximating the greatest height \(H\) reached by the ball is
$$H \approx \frac { U ^ { 2 } } { 2 g } - \frac { \lambda U ^ { 4 } } { 4 m g ^ { 2 } }$$
where \(g\) is the acceleration due to gravity.
Show that this formula is dimensionally consistent.
A better approximation has the form \(H \approx \frac { U ^ { 2 } } { 2 g } - \frac { \lambda U ^ { 4 } } { 4 m g ^ { 2 } } + \frac { 1 } { 6 } \lambda ^ { 2 } U ^ { \alpha } m ^ { \beta } g ^ { \gamma }\).
Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
A girl of mass 50 kg is practising for a bungee jump. She is connected to a fixed point O by a light elastic rope with natural length 24 m and modulus of elasticity 2060 N . At one instant she is 30 m vertically below O and is moving vertically upwards with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She comes to rest instantaneously, with the rope slack, at the point A . Find the distance OA .