Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 6: END-OF-CHAPTER REVIEW EXERCISE 8

A ramp is inclined at an angle ${\sin }^{-1}0.1$ to the horizontal. A box of mass $40 \mathrm{kg}$ is projected up the ramp with initial speed $5 \mathrm{m}{ \mathrm{s}}^{-1}$. The coefficient of friction between the ramp and the box is $0.05$, and no other resistance forces act.

Find the gain in the potential energy of the box.

The total mechanical energy is the sum of the kinetic energy and the potential energy. 

A ramp is inclined at an angle ${\sin }^{-1}0.1$ to the horizontal. A box of mass $40 \mathrm{kg}$ is projected up the ramp with initial speed $5 \mathrm{m}{ \mathrm{s}}^{-1}$. The coefficient of friction between the ramp and the box is $0.05$, and no other resistance forces act.

Show that the overall loss in the mechanical energy of the box is $166 \mathrm{J}$.

Jack has mass $70 \mathrm{kg}$. He works as a 'human cannon ball'. Jack is projected with speed $12 \mathrm{m}{ \mathrm{s}}^{-1}$ at an angle of $45°$ above the horizontal. He lands on a trampoline when the angle between his flight and the horizontal is $50°$. Model Jack as a particle with no air resistance.

Explain why the horizontal component of Jack's velocity is constant.

Jack has mass $70 \mathrm{kg}$. He works as a 'human cannon ball'. Jack is projected with speed $12 \mathrm{m}{ \mathrm{s}}^{-1}$ at an angle of $45°$ above the horizontal. He lands on a trampoline when the angle between his flight and the horizontal is $50°$. Model Jack as a particle with no air resistance.

Find Jack's speed when he hits the trampoline.

Jack has mass $70 \mathrm{kg}$. He works as a 'human cannon ball'. Jack is projected with speed $12 \mathrm{m}{ \mathrm{s}}^{-1}$ at an angle of $45°$ above the horizontal. He lands on a trampoline when the angle between his flight and the horizontal is $50°$. Model Jack as a particle with no air resistance.

Find the kinetic energy gained during the flight.

The gain in Jack's kinetic energy equals the loss in his gravitational potential energy.

Jack has mass $70 \mathrm{kg}$. He works as a 'human cannon ball'. Jack is projected with speed $12 \mathrm{m}{ \mathrm{s}}^{-1}$ at an angle of $45°$ above the horizontal. He lands on a trampoline when the angle between his flight and the horizontal is $50°$. Model Jack as a particle with no air resistance.

Find the difference in height between the mouth of the cannon and the trampoline. (Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

Jack has mass $70 \mathrm{kg}$. He works as a 'human cannon ball'. Jack is projected with speed $12 \mathrm{m}{ \mathrm{s}}^{-1}$ at an angle of $45°$ above the horizontal. He lands on a trampoline when the angle between his flight and the horizontal is $50°$. Model Jack as a particle with no air resistance.

What could happen if $\alpha$ is very small?

Jack has mass $70 \mathrm{kg}$. He works as a 'human cannon ball'. Jack is projected with speed $12 \mathrm{m}{ \mathrm{s}}^{-1}$ at an angle of $45°$ above the horizontal. He lands on a trampoline when the angle between his flight and the horizontal is $50°$. Model Jack as a particle with no air resistance.

By changing the angle of projection, Jack can change the angle between his flight and the horizontal when he lands. Suppose that Jack lands on the trampoline at an angle $\alpha$ to the horizontal.

What could happen if $\alpha$ is close to $90°?$ (Use $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)