91Ó°ÊÓ

A frictionless surface means that no friction forces oppose the motion of objects sliding across it. This is important for the exercise at hand because the girl and the sled move without losing momentum to friction. Friction usually converts some mechanical energy into heat, reducing the objects' acceleration. Here, the absence of friction simplifies calculations, allowing us to focus solely on the kinetic effects of the forces applied. Understanding how a frictionless surface affects motion can illuminate physics principles, like Newton's laws of motion. It helps illustrate pure motion concepts, uninhibited by real-world resistances.

• A frictionless surface allows an object to continue moving indefinitely unless acted upon by an external force.
• Without friction, the motion produced by a force is dictated only by Newton's second law, \( F = ma \), without reducing force effectiveness.
• In this exercise, friction's absence means the girl and sled will accelerate according to the full force exerted on the rope.

Therefore, on a frictionless ice, the girl and sled experience pure motion, quickly highlighting concepts like momentum and acceleration due to applied force.

Acceleration is a critical concept in understanding motion, especially in this exercise. It tells us how swiftly an object speeds up or slows down and is calculated using Newton's second law: \( a = \frac{F}{m} \).

Finding Acceleration of the Sled

Given that the sled has a mass of \( 8.4 \, \mathrm{kg} \) and the force applied through the rope is \( 5.2 \, \mathrm{N} \), we use the formula to find the sled's acceleration:\[a_{\text{sled}} = \frac{5.2}{8.4} = 0.619 \, \mathrm{m/s^2}\]

Calculating the Girl's Acceleration

Similarly, the girl's acceleration can be determined with her mass at \( 40 \, \mathrm{kg} \):\[a_{\text{girl}} = \frac{5.2}{40} = 0.13 \, \mathrm{m/s^2}\]

• Acceleration depends directly on force and inversely on mass. A greater force leads to greater acceleration, while a larger mass results in a smaller one for the same force applied.
• Both the girl and sled accelerate in opposite directions due to the equality of action and reaction forces, even if their accelerations' magnitudes differ.

Understanding acceleration here showcases how mass and force interplay to affect motion differently for the girl and sled.

Relative motion entails understanding how two objects move with respect to one another. In this scenario, both the girl and the sled accelerate towards each other across a frictionless surface. These movements combine to form their relative motion.The initial distance between the girl and the sled is \( 15 \, \mathrm{m} \). To find when and where they meet, we need their relative acceleration:

• The sum of their individual accelerations determines their relative acceleration: \[a_{\text{relative}} = a_{\text{sled}} + a_{\text{girl}} = 0.619 + 0.13 = 0.749 \, \mathrm{m/s^2}\]
• This acceleration tells us how quickly the distance between them decreases.
• Using the equation for motion, \( d = \frac{1}{2} a t^2 \), we can calculate the time it takes for them to meet:\[15 = \frac{1}{2} (0.749) t^2\]\[t^2 = \frac{30}{0.749} \rightarrow t = \sqrt{40.053} \approx 6.33 \, \mathrm{s}\]

Once we know the time, we can find any distances covered numerically and discuss how relative motion interplays. This concept is pivotal because it shows how positions change concerning each other over time, guided by the exerted forces and surface conditions.