A bead is inserted onto a thin line with an inclination \(\alpha\) with the vertical. When the bead is released, its acceleration along the line is \(a=g \cos \alpha\). (a) Find the speed, \(v(t)\), of the bead as a function of time. (b) Find the position \(s(t)\) of the bead along the line as a function of time, \(t\). (c) Find the height, \(h(t)\) of the bead as a function of time, \(t\).

The bead's acceleration on an inclined plane is derived from basic principles of physics. When the bead is released, gravity acts on it, pulling it downward. However, since the plane is inclined, only part of the gravitational force influences the bead's movement along the line. This part is given by the formula:

• \( a = g \cos \alpha \)

where \( g \) is the acceleration due to gravity (approximately \( 9.8 \) meters per second squared). The angle \( \alpha \) represents the inclination of the plane with the vertical. This acceleration describes how fast the bead's velocity changes over time. Once you have the acceleration, you can find out how the speed of the bead increases each second by integration.

Velocity is the rate at which the bead changes its position over time. To find the velocity of the bead (\( v(t) \)), we start from the acceleration formula. Acceleration is the derivative of velocity with respect to time. By integrating the acceleration \( a = g \cos \alpha \) with respect to time, we get:

• \( v(t) = \int{g \cos \alpha \, dt} = g \cos \alpha \, t + C_v \)

Assuming the bead starts from rest (\(t=0 \)), its initial velocity is zero. Therefore, the integration constant \( C_v = 0 \), simplifying the formula to:

• \( v(t) = g \cos \alpha \, t \)

This equation tells us how fast the bead is moving along the inclined plane at any time \( t \).

The position \( s(t) \) describes where the bead is located on the inclined plane at any given time. Velocity is the rate of change of position with respect to time. Starting from the velocity obtained in the previous step ( \( v(t) = g \cos \alpha \, t \)), we integrate to find the position:

• \( s(t) = \int{g \cos \alpha \, t \, dt} = \frac{1}{2} g \cos \alpha \, t^2 + C_s \)

Considering the bead starts at the origin (\( t=0 \) and \( s=0 \)) when time is zero, we get \( C_s = 0 \), giving us:

• \( s(t) = \frac{1}{2} g \cos \alpha \, t^2 \)

This equation shows the exact position of the bead along the length of the inclined plane at any time \( t \).

The height \( h(t) \) refers to how high the bead is above the starting point vertically. To find this, we use the trigonometric relationship between the height and the position on the inclined plane. From the position equation:

• \( s(t) = \frac{1}{2} g \cos \alpha \, t^2 \)

We understand that the height can be obtained through the cosine projection of the position:

• \( h(t) = s(t) \cos \alpha \)
• Substituting \( s(t) \) gives

• \( h(t) = \left(\frac{1}{2} g \cos \alpha \, t^2\right) \cos \alpha = \frac{1}{2} g \cos^2 \alpha \, t^2 \)

This equation helps us determine how the vertical height of the bead changes as time progresses.

Each of the above physical quantities—acceleration, velocity, position, and height—depends on time. These dependencies tell us how the bead's motion evolves. Starting from acceleration, all other quantities are derived step by step with respect to time.

• The acceleration \( a = g \cos \alpha \) remains constant at all times.
• Velocity \( v(t) = g \cos \alpha \, t \) increases linearly with time.
• Position \( s(t) = \frac{1}{2} g \cos \alpha \, t^2 \) follows a quadratic relationship, showing how far the bead has moved from its starting point.
• Height \( h(t) = \frac{1}{2} g \cos^2 \alpha \, t^2 \) also follows a quadratic pattern, representing the bead's vertical height.

By understanding these time-dependent relationships, you can predict the bead's behavior at any given moment.