(a) State two deductions that can be made from a displacement-time graph. (b) If the distance beween two equal masses is doubled and their individual masses...

(a) Deductions from a displacement-time graph

1. The velocity is obtained from the gradient of the graph. For the straight line shown,

\[v=\frac{\Delta s}{\Delta t}=\frac{10-2}{4-0}=2.0\ \text{m s}^{-1}.\]

2. The initial displacement is given by the intercept on the displacement axis. From the graph, the initial displacement is \(2.0\ \text{m}\). A straight line also indicates uniform velocity.

(b) Effect on the gravitational force

The gravitational force between two masses is

\[F=\frac{Gm_1m_2}{r^2}.\]

When each mass and the distance between them are doubled,

\[F' = \frac{G(2m_1)(2m_2)}{(2r)^2}=\frac{4Gm_1m_2}{4r^2}=\frac{Gm_1m_2}{r^2}=F.\]

Therefore, the force between the masses remains unchanged.

(c) Factors affecting the maximum height attained by a fired bullet

1. The initial speed, or muzzle velocity, of the bullet.
2. The angle of projection of the bullet.

Other acceptable factors include acceleration due to gravity and air resistance.

(d) Practical examples of mechanical resonance

1. A glass may shatter when subjected to a sound whose frequency equals the natural frequency of the glass.
2. A bridge may vibrate dangerously when soldiers march across it in step.

Time taken by the body to reach the bottom of the plane

Given: \(h=4.0\ \text{m}\), \(\theta=30^\circ\), \(\mu=0.3\), \(u=0\), and \(g=10\ \text{m s}^{-2}\).

The length \(s\) of the inclined plane is

\[s=\frac{h}{\sin 30^\circ}=\frac{4.0}{0.5}=8.0\ \text{m}.\]

The acceleration down the plane is

\[a=g(\sin\theta-\mu\cos\theta)\]

\[a=10\left(\sin30^\circ-0.3\cos30^\circ\right)\]

\[a=10\left(0.5-0.3\times0.866\right)=2.40\ \text{m s}^{-2}.\]

Using \(s=ut+\tfrac12at^2\),

\[8.0=0+\frac12(2.40)t^2\]

\[t^2=\frac{16.0}{2.40}=6.67\]

\[t=2.58\ \text{s}\approx2.6\ \text{s}.\]

Hence, the body takes \(2.6\ \text{s}\) to reach the bottom of the plane.