The problem you've uploaded describes two particles, $P$ and $Q$, moving along the same horizontal line with constant accelerations. The acceleration of $P$ is $2 \, \text{m/s}^2$, while $Q$ has an acceleration of $3.6 \, \text{m/s}^2$. The motion begins with:

• At time $t = 0$, $P$ passes through point $A$ with a speed of $4 \, \text{m/s}$.
• One second later, $Q$ passes through point $A$ with a speed of $3 \, \text{m/s}$, and both particles are moving in the same direction.

Part (a) asks for expressions of displacement:

You are required to write down expressions for the displacements of $P$ and $Q$ from $A$, in terms of $t$, where $t$ represents the time after $P$ has passed through $A$.

Displacement of particle $P$:

For particle $P$, the displacement after time $t$ can be given by the kinematic equation: $s_P = u_P t + \frac{1}{2} a_P t^2$ where:

• $u_P = 4 \, \text{m/s}$ (initial velocity of $P$),
• $a_P = 2 \, \text{m/s}^2$ (acceleration of $P$),
• $t$ is the time after $P$ has passed through $A$.

Thus, the displacement of $P$ is: $s_P = 4t + \frac{1}{2} \times 2 \times t^2 = 4t + t^2$

Displacement of particle $Q$:

For particle $Q$, since it passes through $A$ one second later, the time elapsed for $Q$ is $t - 1$ (i.e., $t - 1$ seconds have passed for $Q$ when $P$ has traveled for $t$ seconds). The displacement of $Q$ can also be calculated using the kinematic equation: $s_Q = u_Q (t - 1) + \frac{1}{2} a_Q (t - 1)^2$ where:

• $u_Q = 3 \, \text{m/s}$ (initial velocity of $Q$),
• $a_Q = 3.6 \, \text{m/s}^2$ (acceleration of $Q$).

Thus, the displacement of $Q$ is: $s_Q = 3(t - 1) + \frac{1}{2} \times 3.6 \times (t - 1)^2$ Simplifying: $s_Q = 3(t - 1) + 1.8(t - 1)^2$

Summary:

The expressions for the displacements are:

• $s_P = 4t + t^2$
• $s_Q = 3(t - 1) + 1.8(t - 1)^2$

Would you like further clarification or the solution to part (b) of the question?

Relative questions:

1. What would be the displacement of $P$ after 5 seconds?
2. How do the displacements of $P$ and $Q$ compare at $t = 2$ seconds?
3. Can you find the time at which both particles $P$ and $Q$ are at the same position?
4. How would the problem change if the initial speed of $Q$ were greater than $P$?
5. What is the physical significance of the acceleration for each particle in this problem?

Tip:

When solving problems with relative motion, always carefully account for the time difference between the movements of objects.