Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 6: EXERCISE 3D

A ball of mass $3 \mathrm{kg}$ is rolled with initial speed $4 \mathrm{m}{ \mathrm{s}}^{-1}$ up a slope at an angle $10°$ to the horizontal.

Find the maximum distance up the slope the ball reaches.

A ball of mass $3 \mathrm{kg}$ is rolled with initial speed $4 \mathrm{m}{ \mathrm{s}}^{-1}$ up a slope at an angle $10°$ to the horizontal.

What assumptions have been made to answer the question?

A cyclist of mass $70 \mathrm{kg}$ (including her bicycle) arrives at an uphill stretch of road of length $30 \mathrm{m}$ with an angle $9°$ to the horizontal, travelling at $10 \mathrm{m}{ \mathrm{s}}^{-1}.$ She exerts a force of $15 \mathrm{N}$ parallel to the slope and there is wind resistance of $5 \mathrm{N}$ against her. Find the time taken to reach the top of the slope.

A ball of mass $m \mathrm{kg}$ is rolled up a slope at an angle $\theta$ to the horizontal, where $\sin \theta =\frac{2}{5}.$ The ball passes a point $A$ with speed $7 \mathrm{m}{ \mathrm{s}}^{-1}.$ A point $B$ is $5 \mathrm{m}$ further up the slope than point $A.$ Find the time between passing $B$ on the way up and returning to $B$ on the way down.

A van of mass $2000 \mathrm{kg}$ is towed from rest by two ropes. One pulls with a tension of $130 \mathrm{N}$ at $10°$ to the direction of motion and the other acts at $15°$ to the direction of motion. Find the distance covered in $10 \mathrm{s}.$

A ship of mass $15000 \mathrm{kg}$ is moving due east at $2 \mathrm{m}{ \mathrm{s}}^{-1}$ when it starts being towed by a tugboat. The wind is blowing it on a bearing of $60°,$ so the tugboat exerts a force of $5000 \mathrm{N}$ on a bearing of $100°$ to make the ship continue to go east. Find the speed of the ship after $5 \mathrm{s}.$

A box of mass $9 \mathrm{kg}$ is dragged along horizontal ground by a force $F$ acting at $30°$ above the horizontal. There is friction of $5 \mathrm{N}.$ The box starts at rest and reaches a speed of $4 \mathrm{m}{ \mathrm{s}}^{-1}$ in $10 \mathrm{m}.$ Find the size of the force $F.$

A car of mass $1200 \mathrm{kg}$ arrives at a steep upwards slope of length $130 \mathrm{m}$ at $34°$ to the horizontal. It is travelling at $12 \mathrm{m}{ \mathrm{s}}^{-1}$ and there is air resistance of $100 \mathrm{N}.$ Find the minimum force, assumed constant, the engine must provide for the car to reach the top of the slope.