91Ó°ÊÓ

Linear acceleration refers to the change in velocity over time for an object moving along a straight path. In our exercise, the cyclist begins from a standstill and speeds up at an acceleration rate of \(1.00 \, \text{m/s}^2\). The formula for determining velocity from acceleration is \(v = u + at\), where \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time over which acceleration occurs. In this case, the cyclist's initial velocity \(u\) is zero because they start from rest.

Therefore, after applying the known values: acceleration \(a = 1.00 \, \text{m/s}^2\) and time \(t = 2.5 \, \text{s}\), we find the velocity as:

• Velocity \(v = 0 + (1.00 \, \text{m/s}^2 \times 2.5 \, \text{s}) = 2.5 \, \text{m/s}\).

This result gives us the linear speed of the bicycle after 2.5 seconds.

Rotational motion occurs when an object spins around an axis. In the case of a bicycle tire, the tire rotates as the bicycle moves forward. The unique aspect of rotational motion is how it combines with linear motion when the tire rolls on the ground.

Each point on the tire traces out a unique path. While the center of the tire moves straight forward at the same speed as the bicycle, the rim of the tire moves in a more complex manner. Specifically, at any given time:

• The bottom of the tire, where it touches the ground, is momentarily at rest because it's in contact with the ground.
• The top of the tire moves at a speed greater than the center due to rotational effects.

This combined motion means that the speed at the highest point on the tire, or the point at the top of the rim, can be particularly fascinating. As shown in our exercise, this top point moves at twice the speed of the bicycle’s center, showing how linear and rotational motions interact.

Calculating velocity in the context of combined linear and rotational motion requires understanding how these two types of motion contribute to the overall speed. It’s not just about how fast the bike itself is moving in a straight line, but also how the rotation of its wheels adds to this speed.

Let’s break this down:

• The linear speed \(v\) of the bicycle, calculated from the formula \(v = u + at\), is \(2.5 \, \text{m/s}\) after \(2.5 \, \text{s}\).
• The rotational motion means the top of the wheel moves faster than the rest of the tire.

In our specific scenario with a 68 cm diameter tire, the speed at the top point of the rim is twice the center speed. Therefore, the top of the tire moves at \(v_{\text{top}} = 2 \times 2.5 \, \text{m/s} = 5.0 \, \text{m/s}\). This calculation demonstrates the interplay of linear and rotational dynamics, revealing the full velocity at different points on the bicycle tire.