SPS SPS SM Mechanics (SPS SM Mechanics) 2021 September

1. A racing car starts from rest at the point \(A\) and moves with constant acceleration.of \(11 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 8 s . The velocity it has reached after 8 s is then maintained for \(T \mathrm {~s}\). The racing car then decelerates from this velocity to \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a further 2 s , reaching point \(B\).
   a Sketch a velocity-time graph to illustrate the motion of the racing car. Include the top speed of the racing car in your sketch.
   b Given that the distance between \(A\) and \(B\) is 1404 m , find the value of \(T\).
   [0pt] [BLANK PAGE]
2. A particle \(P\) is acted upon by three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) given by \(\mathbf { F } _ { 1 } = ( 6 \mathbf { i } - 4 \mathbf { j } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 3 } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }\), where \(a\) and \(b\) are constants. Given that \(P\) is in equilibrium,
   a find the value of \(a\) and the value of \(b\).

The force \(\mathbf { F } _ { 2 }\) is now removed. The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 3 }\) is \(\mathbf { R }\).
b Find the magnitude of \(\mathbf { R }\).
c Find the angle, to \(0.1 ^ { \circ }\), that \(\mathbf { R }\) makes with \(\mathbf { i }\).
[0pt] [BLANK PAGE]

3. A car of mass 1200 kg pulls a trailer of mass 400 kg along a straight horizontal road. The car and trailer are connected by a tow-rope modelled as a light inextensible rod. The engine of the car provides a constant driving force of 3200 N . The horizontal resistances of the car and the trailer are proportional to their respective masses. Given that the acceleration of the car and the trailer is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
a find the resistance to motion on the trailer,
b find the tension in the tow-rope. When the car and trailer are travelling at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the tow-rope breaks. Assuming that the resistances to motion remain unchanged,
c find the distance the trailer travels before coming to a stop,
d state how you have used the modelling assumption that the tow-rope is inextensible.
[0pt] [BLANK PAGE]

4. A car starts from the point \(A\). At time \(t \mathrm {~s}\) after leaving \(A\), the distance of the car from \(A\) is \(s \mathrm {~m}\), where \(s = 30 t - 0.4 t ^ { 2 } , 0 \leqslant t \leqslant 25\). The car reaches the point \(B\) when \(t = 25\).
a Find the distance \(A B\).
b Show that the car travels with a constant acceleration and state the value of this acceleration. A runner passes through \(B\) when \(t = 0\) with an initial velocity of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) running directly towards \(A\). The runner has a constant acceleration of \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
c Find the distance from \(A\) at which the runner and the car pass one another.
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE] End of examination