Understanding Car Motion from Speed Data

The question provides us with the speed of a car recorded at successive 1-second intervals and asks us to evaluate several statements about its motion. We need to analyze the given time and speed data to determine which statements are correct.

Analyzing the Car Speed Data

Let's first look at the data provided:

From this table, we can see how the car's speed changes over time.

Evaluating Statement 1: Uniform Acceleration

The first statement says the car travels with a uniform acceleration of 2 m/s². Acceleration is the rate of change of speed. To check if the acceleration is uniform, we calculate the change in speed over each 1-second interval:

• From t = 0 s to t = 1 s: Change in speed = \(2 \, \text{m/s} - 0 \, \text{m/s} = 2 \, \text{m/s}\). Acceleration = \(\frac{\text{Change in speed}}{\text{Time interval}} = \frac{2 \, \text{m/s}}{1 \, \text{s}} = 2 \, \text{m/s}^2\).
• From t = 1 s to t = 2 s: Change in speed = \(4 \, \text{m/s} - 2 \, \text{m/s} = 2 \, \text{m/s}\). Acceleration = \(\frac{2 \, \text{m/s}}{1 \, \text{s}} = 2 \, \text{m/s}^2\).
• From t = 2 s to t = 3 s: Change in speed = \(6 \, \text{m/s} - 4 \, \text{m/s} = 2 \, \text{m/s}\). Acceleration = \(\frac{2 \, \text{m/s}}{1 \, \text{s}} = 2 \, \text{m/s}^2\).
• From t = 3 s to t = 4 s: Change in speed = \(8 \, \text{m/s} - 6 \, \text{m/s} = 2 \, \text{m/s}\). Acceleration = \(\frac{2 \, \text{m/s}}{1 \, \text{s}} = 2 \, \text{m/s}^2\).

Since the acceleration is constant at 2 m/s² over each interval, the car has a uniform acceleration of 2 m/s². So, statement 1 is correct.

Evaluating Statement 2: Distance Traveled in 4 Seconds

The second statement says the car travels 16 m in 4 s. Since the car starts from rest (speed is 0 m/s at t=0) and moves with uniform acceleration, we can use the kinematic equation for displacement:

\[ s = ut + \frac{1}{2}at^2 \]where:

• \(s\) is the displacement (distance traveled in this case, as it's a straight road with no change in direction)
• \(u\) is the initial speed (at t=0)
• \(a\) is the uniform acceleration
• \(t\) is the time taken

From the data and statement 1, we know:

• Initial speed, \(u = 0 \, \text{m/s}\)
• Acceleration, \(a = 2 \, \text{m/s}^2\)
• Time, \(t = 4 \, \text{s}\)

Plugging these values into the equation:

\[ s = (0 \, \text{m/s})(4 \, \text{s}) + \frac{1}{2}(2 \, \text{m/s}^2)(4 \, \text{s})^2 \]

\[ s = 0 + \frac{1}{2}(2)(16) \, \text{m} \]

\[ s = 16 \, \text{m} \]

So, the car travels 16 m in 4 seconds. Statement 2 is correct.

Evaluating Statement 3: Average Speed

The third statement says the car travels with an average speed of 4 m/s. Average speed is defined as the total distance traveled divided by the total time taken.

• Total distance traveled = 16 m (from statement 2 analysis)
• Total time taken = 4 s

\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{16 \, \text{m}}{4 \, \text{s}} = 4 \, \text{m/s} \]

Alternatively, for motion with uniform acceleration, the average speed over a time interval is also the average of the initial and final speeds during that interval.

• Initial speed (at t=0) = 0 m/s
• Final speed (at t=4 s) = 8 m/s

\[ \text{Average speed} = \frac{\text{Initial speed} + \text{Final speed}}{2} = \frac{0 \, \text{m/s} + 8 \, \text{m/s}}{2} = \frac{8 \, \text{m/s}}{2} = 4 \, \text{m/s} \]

So, the average speed of the car over 4 seconds is 4 m/s. Statement 3 is correct.

Conclusion on Car Motion Statements

Based on our analysis, all three statements are correct:

1. The car travels with a uniform acceleration of 2 m/s².
2. The car travels 16 m in 4 s.
3. The car travels with an average speed of 4 m/s.

Therefore, the correct option should include all three statements.

Revision Table: Kinematics Concepts

Additional Information: Uniformly Accelerated Motion

When an object moves with uniform acceleration, its speed changes by the same amount in every equal interval of time. This type of motion is governed by a set of equations called kinematic equations, which relate displacement (\(s\)), initial velocity (\(u\)), final velocity (\(v\)), acceleration (\(a\)), and time (\(t\)). The equations used in this solution are derived from these principles. Understanding uniform acceleration is fundamental to solving many problems in mechanics.