91Ó°ÊÓ

The equations of motion are essential tools in kinematics, especially when dealing with uniformly accelerated motion. They help us predict the future position and velocity of an object. In this exercise, we used the basic equation of motion:

• \( x = x_0 + v_0 t + \frac{1}{2} a t^2 \)

Here:

• \(x\) is the final position.
• \(x_0\) is the initial position.
• \(v_0\) is the initial velocity; for both vehicles, it was 0 since they started from rest.
• \(a\) represents the acceleration.
• \(t\) is the time elapsed.

By substituting the specific values for the truck and automobile, we establishes their motion equations. This step set the foundation for finding when the automobile overtakes the truck and calculates other unknowns.

Uniform acceleration means that an object's velocity changes at a constant rate. It is a straightforward but vital concept in physics that simplifies many calculations. When an object experiences uniform acceleration, it follows the equations of motion without any change in the acceleration term.
In our scenario, both the truck and the automobile start with different uniform accelerations:

• The truck has an acceleration of \(2.10 \text{ m/s}^2\).
• The automobile has a greater acceleration of \(3.40 \text{ m/s}^2\).

This difference in acceleration is crucial because despite starting behind, the automobile's higher acceleration allows it to catch up to the truck. Uniform acceleration ensures that the motion is predictable and can be neatly expressed with the equations of motion, aiding in the determination of when and where the vehicles meet.

Velocity calculation is necessary to understand how fast each vehicle is moving at the moment they meet. Using the formula:

• \( v = v_0 + at \)

We can calculate the final velocities for both vehicles. Since both started from rest, \(v_0 = 0\), making our calculations straightforward:

• For the truck: \( v_{\text{truck}} = 2.10 \times 6.17 = 12.957 \text{ m/s} \)
• For the automobile: \( v_{\text{auto}} = 3.40 \times 6.17 = 20.978 \text{ m/s} \)

These calculations show that when the automobile catches up to the truck, its speed is significantly higher. This higher speed is a result of its greater uniform acceleration over the same period.

A position-time graph visually represents how the position of an object changes over time. In the context of this exercise, it shows the intersection point where the automobile overtakes the truck. To sketch the graph:

• The truck's position as a function of time is given by \( x_{\text{truck}} = 1.05t^2 \).
• The automobile's position is \( x_{\text{auto}} = -24.60 + 1.70t^2 \).

The graph should show:

• Two parabolic curves originating from their respective initial positions.
• The truck's curve starting at the origin and the automobile's curve starting from a negative position, moving upwards.
• The point of intersection occurs at \( t \approx 6.17 \) seconds and \( x = 40 \) meters, indicating where both vehicles align spatially.

This visual tool not only helps understand the problem better but also allows us to verify the results obtained from mathematical calculations.