3.02d Constant acceleration: SUVAT formulae

A particle of mass \(m\) moves in a straight line with constant acceleration \(a\). Its initial and final velocities are \(u\) and \(v\) respectively and its final displacement from its starting position is \(s\). In order to model the motion of the particle it is suggested that the velocity is given by the equation $$v^2 = pu^{\alpha} + qa^{\beta}s^{\gamma}$$ where \(p\) and \(q\) are dimensionless constants.

1. Explain why \(\alpha\) must equal 2 for the equation to be dimensionally consistent. [2]
2. By using dimensional analysis, determine the values of \(\beta\) and \(\gamma\). [4]
3. By considering the case where \(s = 0\), determine the value of \(p\). [1]
4. By multiplying both sides of the equation by \(\frac{1}{2}m\), and using the numerical values of \(\alpha\), \(\beta\) and \(\gamma\), determine the value of \(q\). [2]

A jogger is running along a straight horizontal road. The jogger starts from rest and accelerates at a constant rate of \(0.4\,\text{m}\,\text{s}^{-2}\) until reaching a velocity of \(V\,\text{m}\,\text{s}^{-1}\). The jogger then runs at a constant velocity of \(V\,\text{m}\,\text{s}^{-1}\) before decelerating at a constant rate of \(0.08\,\text{m}\,\text{s}^{-2}\) back to rest. The jogger runs a total distance of \(880\,\text{m}\) in \(250\,\text{s}\).

1. Sketch the velocity-time graph for the jogger's journey. [2]
2. Show that \(3V^2 - 100V + 352 = 0\). [6]
3. Hence find the value of \(V\), giving a reason for your answer. [3]

Particle \(A\), of mass \(m\) kg, lies on the plane \(\Pi_1\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. Particle \(B\), of \(4m\) kg, lies on the plane \(\Pi_2\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at \(P\). The coefficient of friction between particle \(A\) and \(\Pi_1\) is \(\frac{1}{4}\) and plane \(\Pi_2\) is smooth. Particle \(A\) is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane. This is shown on the diagram below. \includegraphics{figure_13}

1. Show that when \(A\) is released it accelerates towards the pulley at \(\frac{7g}{15}\) m s\(^{-2}\). [6]
2. Assuming that \(A\) does not reach the pulley, show that it has moved a distance of \(\frac{1}{4}\) m when its speed is \(\sqrt{\frac{7g}{30}}\) m s\(^{-1}\). [2]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. Distance is measured in metres and time in seconds. A ship of mass 100 000 kg is being towed by two tug boats. • The cables attaching each tug to the ship are horizontal. • One tug produces a force of \((350\mathbf{i} + 400\mathbf{j})\) N. • The other tug produces a force of \((250\mathbf{i} - 400\mathbf{j})\) N. • The total resistance to motion is 200 N. • At the instant when the tugs begin to tow the ship, it is moving east at a speed of 1.5 m s\(^{-1}\).

1. Explain why the ship continues to move directly east. [2]
2. Find the acceleration of the ship. [2]
3. Find the time which the ship takes to move 400 m while it is being towed. Find its speed after moving that distance. [6]

A particle is projected from a point \(P\) on an inclined plane, up the line of greatest slope through \(P\), with initial speed \(V\). The angle of the plane to the horizontal is \(\theta\).

1. If the plane is smooth, and the particle travels for a time \(\frac{2V}{g}\cos\theta\) before coming instantaneously to rest, show that \(\theta = \frac{1}{4}\pi\). [4]
2. If the same plane is given a roughened surface, with a coefficient of friction 0.5, find the distance travelled before the particle comes instantaneously to rest. [5]

10 seconds after passing a warning signal, a train is travelling at 18 m s\(^{-1}\) and has gone 215 m beyond the signal. Find the acceleration (assumed to be constant) of the train during the 10 seconds and its velocity as it passed the signal. [6]

A stone is projected vertically upwards from ground level at a speed of \(30\,\mathrm{m}\,\mathrm{s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]

An engine is travelling along a straight horizontal track against negligible resistances. In travelling a distance of 750 m its speed increases from 5 m s\(^{-1}\) to 15 m s\(^{-1}\). Find the time taken if the engine was

1. exerting a constant tractive force, [2]
2. working at constant power. [9]

A stone is projected vertically upwards from ground level at a speed of \(30 \mathrm{~m} \mathrm{~s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]

A particle \(P\) is free to move along a straight line \(Ox\). It starts from rest at \(O\) and after \(t\) seconds its acceleration \(a \mathrm{~m} \mathrm{~s}^{-2}\) is given by \(a = 12 - 6t\).

1. Find an expression in terms of \(t\) for its velocity \(v \mathrm{~m} \mathrm{~s}^{-1}\). Hence find the velocity of \(P\) when \(t = 4\). [4]
2. Find the displacement of \(P\) from \(O\) when \(t = 4\). [3]
3. Find the velocity of \(P\) when it returns to \(O\). [3]

A stone is projected vertically upwards from ground level at a speed of \(30 \mathrm{m} \mathrm{s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]

A particle \(P\) is free to move along a straight line \(Ox\). It starts from rest at \(O\) and after \(t\) seconds its acceleration \(a \mathrm{m} \mathrm{s}^{-2}\) is given by \(a = 12 - 6t\).

1. Find an expression in terms of \(t\) for its velocity \(v \mathrm{m} \mathrm{s}^{-1}\). Hence find the velocity of \(P\) when \(t = 4\). [4]
2. Find the displacement of \(P\) from \(O\) when \(t = 4\). [3]
3. Find the velocity of \(P\) when it returns to \(O\). [3]

A particle is projected with speed \(U \text{ m s}^{-1}\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac{12}{13}\), and reaches its maximum height after \(2.4\) seconds.

1. Find \(U\) and the maximum height reached by the particle. [4]
2. Find the horizontal range of the particle. [4]

A particle \(P\) of mass \(2\) kg rests on a long rough horizontal table. The coefficient of friction between \(P\) and the table is \(0.2\). A light inextensible string has one end attached to \(P\) and the other end attached to a particle \(Q\) of mass \(4\) kg. The particle \(Q\) is placed on a smooth plane inclined at \(30^{\circ}\) to the horizontal. The string passes over a smooth light pulley fixed at a point in the line of intersection of the table and the plane (see diagram). \includegraphics{figure_12} Initially the system is held in equilibrium with the string taut. The system is released from rest at time \(t = 0\), where \(t\) is measured in seconds. In the subsequent motion \(P\) does not reach the pulley.

1. Show that the magnitude of the acceleration of the particles is \(\frac{8}{3}\) m s\(^{-2}\). [4]

After the particles have moved a distance of \(12\) m the string is cut.

1. Find the corresponding value of \(t\) and the speed of the particles at this instant. [4]
2. Find the value of \(t\) when \(P\) comes to rest. [3]

4 A particle \(P\) is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground.

1. Find the height of \(P\) above the ground when \(P\) has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Calculate the length of time for which the speed of \(P\) is less than \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and find the horizontal distance travelled by \(P\) during this time.

3 A ball is projected horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a tower which is 30 m high. The tower stands on horizontal ground.

1. Find the speed and direction of motion of the ball when it reaches the ground.
2. Calculate the distance from the foot of the tower to the point where the ball reaches the ground.