A car accelerates from rest at a constant rate of 2 m/s² for some time. Then, it retards at a constant rate of 4 m/s² and comes to rest. What is the maximum speed attained by the car, if it remains in motion for 3 seconds?

To find the maximum speed attained by the car, we need to break down the problem into two phases: the acceleration phase and the deceleration phase. The car accelerates from rest for a certain duration and then decelerates until it comes to a stop. Given that the total time of motion is 3 seconds, we can calculate the maximum speed using the provided acceleration and deceleration rates.

Phase 1: Acceleration

The car starts from rest and accelerates at a constant rate of 2 m/s². We can use the formula for velocity under constant acceleration:

v = u + at

Where:

• v = final velocity
• u = initial velocity (0 m/s, since it starts from rest)
• a = acceleration (2 m/s²)
• t = time of acceleration

Let’s denote the time spent accelerating as t₁. The maximum speed attained at the end of this phase will be:

v = 0 + (2 m/s²) * t₁ = 2t₁

Phase 2: Deceleration

After reaching the maximum speed, the car begins to decelerate at a rate of 4 m/s². The time spent in this phase will be t₂. Since the total time of motion is 3 seconds, we can express this as:

t₁ + t₂ = 3 seconds

During deceleration, the final velocity will be 0 m/s when the car comes to rest. We can again use the formula for velocity:

0 = v - (4 m/s²) * t₂

Substituting the expression for v from the acceleration phase:

0 = 2t₁ - (4 m/s²) * t₂

Setting Up the Equations

Now we have two equations:

• t₁ + t₂ = 3
• 2t₁ = 4t₂

From the second equation, we can express t₂ in terms of t₁:

t₂ = (2/4)t₁ = (1/2)t₁

Substituting Values

Now, substitute t₂ back into the first equation:

t₁ + (1/2)t₁ = 3

This simplifies to:

(3/2)t₁ = 3

Solving for t₁ gives:

t₁ = 3 / (3/2) = 2 seconds

Now, substituting t₁ back to find t₂:

t₂ = 3 - t₁ = 3 - 2 = 1 second

Calculating Maximum Speed

Now that we have t₁, we can find the maximum speed:

v = 2t₁ = 2 * 2 = 4 m/s

Final Result

The maximum speed attained by the car during its motion is 4 m/s. This calculation illustrates how the phases of motion can be analyzed separately to derive the overall behavior of the car.