https://file.aiquickdraw.com/mathbot/file/1726474374108utqzhhp9.jpg

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.

Two particles P and Q are moving along the same straight horizontal line with constant accelerations 2 m/s² and 3.6 m/s² respectively. At time t = 0, P passes through point A with a speed of 4 m/s. One second later, Q passes through A with a speed of 3 m/s, moving in the same direction as P. Write down expressions for the displacements of P and Q from A, in terms of t, where t seconds is the time after P has passed through A.

In this kinematics problem, learn how to find the displacement of two particles, P and Q, moving along a horizontal line with constant accelerations of 2 m/s² and 3.6 m/s², respectively. We derive the expressions for the displacements of P and Q from point A in terms of time t. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation