91Ó°ÊÓ

Acceleration is one of the fundamental concepts in kinematics, which is a branch of physics dealing with the motion of objects. In our example, the sprinter exerts a force on the ground, which propels him forward. To find out how quickly his speed is increasing, we calculate the acceleration.

Using Newton's second law of motion, which states that force equals mass times acceleration (\(F = m \times a\)), we can rearrange the formula to solve for acceleration (\(a = \frac{F}{m}\)). This equation essentially tells us how much the velocity of an object changes over time due to applied forces.

In our scenario:

• The force exerted (\(F\)) is 650 N.
• The mass of the sprinter (\(m\)) is 70.0 kg.
• Substituting these values into the formula gives us the acceleration: \(a = \frac{650\ N}{70.0\ kg}\), which results in approximately 9.29 \(\mathrm{m/s^2}\).

Understanding acceleration is crucial, not just in sports physics, but in a wide array of practical and theoretical applications ranging from car safety designs to space travel.

Calculating the final speed of an object is integral to understanding its motion. The final speed is essentially the velocity of the object after a certain period of accelerated motion. In kinematics, we often use the equation \(v_f = v_i + a \times t\), where \(v_f\) is the final speed, \(v_i\) is the initial speed, \(a\) is acceleration, and \(t\) is time.

For the sprinter starting from rest, the initial speed, \(v_i\), is 0. Therefore, the equation simplifies to \(v_f = a \times t\). This shows directly how acceleration over a given time increases speed.

After calculating the acceleration as 9.29 \(\mathrm{m/s^2}\), and given that the force is applied for 0.800 seconds, the sprinter's final speed is found by multiplying these two values, yielding approximately 7.43 \(\mathrm{m/s}\).

This calculation of final speed is essential in various scenarios, like determining the required runway length for airplanes to take off, or the stopping distance required when braking a vehicle.

Finally, calculating the distance traveled during acceleration ties together the aspects of kinematics we've been exploring. To find this distance, we can use the kinematic equation \(d = v_i \times t + \frac{1}{2} \times a \times t^2\), which accounts for initial speed, acceleration, and time.

In cases where an object starts from rest, like our sprinter, the initial speed is zero, simplifying the equation to \(d = \frac{1}{2} \times a \times t^2\). This part of the equation, \(\frac{1}{2} \times a \times t^2\), represents the distance covered due to acceleration.

For our sprinter, with an acceleration of 9.29 \(\mathrm{m/s^2}\) and a time interval of 0.800 seconds, we can determine that he travels roughly 2.97 meters. This calculation is pivotal in areas such as determining the length of the run-up for high jumpers, or the safety zones behind stop signs at road intersections.

It's important to approach these kinematic calculations methodically to ensure a solid grasp of how objects move in our world, illuminating more complex systems and fuels scientific inquiry.