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2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2
4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6
- 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt]
[1 mark]
\(\left[ \begin{array} { l } 4
1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 4
3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8
2 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
\(\left[ \begin{array} { l } 8
6 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level.
3
- Calculate the initial kinetic energy of the ball.
3
- The maximum height of the ball above the ground is \(h\) metres.
Jeff and Gurjas use an energy method to find \(h\)
Jeff concludes that \(h = 7.3\)
Gurjas concludes that \(h < 7.3\)
Explain the reasoning that they have used, showing any calculations that you make.
4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave.
Surfers classify waves into one of several types related to their wavelengths.
Two of these classifications are deep water waves and shallow water waves.
4 - The wavelength \(w\) of a deep water wave is given by
$$w = \frac { g t ^ { 2 } } { k }$$
where \(g\) is the acceleration due to gravity and \(t\) is the time period between consecutive waves.
Given that the formula for a deep water wave is dimensionally consistent, show that \(k\) is a dimensionless constant.
4
- The wavelength \(w\) of a shallow water wave is given by
$$w = ( g d ) ^ { \alpha } t ^ { \beta }$$
where \(g\) is the acceleration due to gravity, \(d\) is the depth of water and \(t\) is the time period between consecutive waves.
Use dimensional analysis to find the values of \(\alpha\) and \(\beta\)
5 A car, of mass 1000 kg , is travelling on a straight horizontal road.
When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons.
The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
Find the maximum power output of the car.
Fully justify your answer.