Question

A runner of mass 56.1 kg starts from rest and accelerates with a constant acceleration of \(1.23 \mathrm{~m} / \mathrm{s}^{2}\) until she reaches a velocity of \(5.10 \mathrm{~m} / \mathrm{s}\). She then continues running at this constant velocity. How long does the runner take to travel \(173 \mathrm{~m} ?\)

Short answer

Answer: The runner takes approximately 35.96 seconds to travel 173 meters.

Step-by-step solution

Identify the given values and the equations needed.

The given values are: - Mass of the runner (m): 56.1 kg - Initial velocity (u): 0 m/s (starts from rest) - Acceleration (a): 1.23 m/s² - Final velocity (v): 5.10 m/s - Total distance (s): 173 m The equations of motion we can use for this problem are: 1. v = u + at 2. s = ut + 0.5(at²)

Calculate the time taken to accelerate to the final velocity.

Using the first equation of motion: v = u + at, we can find the time taken (t1) to accelerate to the final velocity. v = 5.10 m/s u = 0 m/s a = 1.23 m/s² 5.10 = 0 + (1.23 × t1) Rearranging the equation to solve for t1: t1 = 5.10 ÷ 1.23 ≈ 4.15 s

Calculate the distance covered during acceleration.

Now, using the second equation of motion, we can find the distance (s1) covered during acceleration: s1 = ut + 0.5(at²) s1 = 0 × 4.15 + 0.5(1.23 × (4.15)²) ≈ 10.72 m

Calculate the remaining distance and time taken at constant velocity.

Subtract the distance covered during acceleration (s1) from the total distance (s) to find the remaining distance (s2) that the runner completes at constant velocity: s2 = s - s1 = 173 - 10.72 ≈ 162.28 m Now using the formula: s = vt, we can find the time taken (t2) to cover the remaining distance at constant velocity: s2 = 162.28 m v = 5.10 m/s t2 = 162.28 ÷ 5.10 ≈ 31.81 s

Calculate the total time taken.

Finally, we can find the total time taken by the runner to travel 173 meters by adding the time taken during acceleration (t1) and the time taken at constant velocity (t2): Total time = t1 + t2 = 4.15 + 31.81 ≈ 35.96 s Thus, the runner takes approximately 35.96 seconds to travel 173 meters.

Key concepts

Equations of Motion

Understanding the equations of motion is crucial for solving kinematics problems involving objects moving with uniform acceleration. These equations describe the relationship between an object's displacement, initial and final velocities, acceleration, and time.

• The first equation v = u + at connects the final velocity (v), initial velocity (u), acceleration (a), and time (t). This tells us how an object’s velocity changes with time under constant acceleration.
• The second equation s = ut + 0.5at² relates displacement (s) to initial velocity, acceleration, and time. It gives us the distance an object has traveled over a period.

These formulae allow us to calculate different parameters of an object's motion when a certain amount of information is provided. In the given problem, we use these equations to find out how long it takes for a runner to accelerate to a specific speed and the distance covered during that acceleration.

Constant Acceleration

The concept of constant acceleration is key to simplifying many physics problems. When an object is said to accelerate at a constant rate, it means that its velocity is increasing by an equal amount each second. In our problem, the runner accelerates with a fixed acceleration for a certain time.

For motion under a constant acceleration, we can apply the kinematic equations universally without complex calculus. It is important to remember this scenario assumes a simplified model where external factors such as friction or air resistance are neglected.When the acceleration remains steady over time, the motion can be easily predicted and calculated using the kinematic equations, which are derived assuming a constant acceleration environment.

Velocity-Time Relationship

The relationship between velocity and time in the context of uniform acceleration is portrayed by the equation v = u + at. This linear relationship is graphically represented by a straight line on a velocity-time graph, where the slope of the line corresponds to the acceleration.

The velocity-time relationship not only tells us how fast an object is moving at a given point in time but also allows us to calculate the distance travelled through the area under the graph. In the runner's scenario, we determined the time it took her to reach a certain velocity. Continuing with a constant velocity implies that there will be no further change in speed, hence no additional acceleration. The time spent running at constant velocity can be calculated by simply dividing the distance by velocity (s = vt).