6 A car of mass 1800 kg is towing a trailer of mass 300 kg up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). The car and trailer are connected by a tow-bar which is light and rigid and is parallel to the road. There is a resistance force of 800 N acting on the car and a resistance force of \(F \mathrm {~N}\) acting on the trailer. The driving force of the car's engine is 3000 N .
- It is given that \(F = 100\).
Find the acceleration of the car and the tension in the tow-bar.
- It is given instead that the total work done against \(F\) in moving a distance of 50 m up the road is 6000 J . The speed of the car at the start of the 50 m is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Use an energy method to find the speed of the car at the end of the 50 m .
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The diagram shows two particles \(P\) and \(Q\) which lie on a line of greatest slope of a plane \(A B C\). Particles \(P\) and \(Q\) are each of mass \(m \mathrm {~kg}\). The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.6\). The length of \(A B\) is 0.75 m and the length of \(B C\) is 3.25 m . The section \(A B\) of the plane is smooth and the section \(B C\) is rough. The coefficient of friction between each particle and the section \(B C\) is 0.25 . Particle \(P\) is released from rest at \(A\). At the same instant, particle \(Q\) is released from rest at \(B\). - Verify that particle \(P\) reaches \(B 0.5 \mathrm {~s}\) after it is released, with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Find the time that it takes from the instant the two particles are released until they collide.
The two particles coalesce when they collide. The coefficient of friction between the combined particle and the plane is still 0.25 . - Find the time that it takes from the instant the particles collide until the combined particle reaches \(C\).
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