Problem 43 - Calculus: Early Transcendentals

Question

A particle moves along a straight line with equation of motion \( s = f(t) \), where \( s \) is measured in meters and \( t \) in seconds. Find the velocity and the speed when \( t = 4 \). \( f(t) = 80t - 6t^2 \)

Step-by-Step Solution

Answer

The velocity is 32 m/s and the speed is 32 m/s at \( t = 4 \).

Step 1: Define Velocity Function

The velocity of a particle is the derivative of its position function with respect to time. The position function is given by \( f(t) = 80t - 6t^2 \). To find the velocity, we need to differentiate this function with respect to \( t \).

Step 2: Differentiate to Find Velocity

Differentiate \( f(t) = 80t - 6t^2 \) to get \( v(t) = f'(t) \). Calculating the derivative, \( f'(t) = 80 - 12t \). This is the velocity function \( v(t) \).

Step 3: Calculate Velocity at t = 4

Substitute \( t = 4 \) into the velocity function \( v(t) = 80 - 12t \) to find the specific velocity. \( v(4) = 80 - 12 \times 4 = 80 - 48 = 32 \). Thus, the velocity of the particle when \( t = 4 \) is 32 meters per second.

Step 4: Determine the Speed

Speed is the absolute value of velocity. Since the velocity at \( t = 4 \) is 32 m/s, the speed is \( |32| = 32 \) m/s.

Key Concepts

Derivative

In calculus, a derivative represents the rate at which a quantity changes. This is a fundamental concept, particularly when dealing with changes in physical quantities like position or velocity over time. To find the derivative of a function, we essentially calculate the slope of the tangent line to the curve represented by that function at any given point. For our exercise, you will use the position function of the particle, given by \( f(t) = 80t - 6t^2 \). This function depends on time, \( t \), and gives the position, \( s \), at any time \( t \).

• To find the derivative, you need to apply basic rules of differentiation. Generally, if you have a polynomial function such as \( ax^n \), its derivative is \( nax^{n-1} \).

By differentiating \( f(t) \), we get the velocity function \( v(t) = f'(t) = 80 - 12t \). This means that at any time \( t \), the rate at which the particle's position is changing (its velocity) is given by this new function.

Velocity

Velocity is an essential concept that describes how fast an object moves and in which direction. It isn't just about how speedily something is moving; it also considers which way it's going, making it a vector quantity. In simpler terms, velocity tells us both the rate of movement and its direction.

• In the given exercise, velocity is derived from the position function \( f(t) = 80t - 6t^2 \). We found that the velocity function is \( v(t) = 80 - 12t \).
• This velocity function allows us to determine the precise velocity of the particle at any particular time \( t \).

For instance, when the time \( t = 4 \), by substituting into the velocity function, we obtain \( v(4) = 32 \) meters per second. This value shows how fast and in which direction the particle is moving at exactly \( t = 4 \) seconds.

Speed

Speed is a scalar quantity that indicates how fast an object is moving, regardless of its direction. Unlike velocity, it doesn't tell you anything about direction, only the magnitude of movement. In short, speed measures the rate of distance traveled per unit of time, without considering where the movement is headed.In our problem:

• The speed at any moment is the absolute value of velocity.
• This is because speed only concerns itself with how fast the object is moving, not the direction.

When the velocity of the particle at \( t = 4 \) was calculated to be 32 meters per second, the speed was simply the absolute value \( |32| = 32 \) m/s. This is straightforward because the direction doesn't affect speed; only the magnitude matters.

Equation of Motion

An equation of motion describes how the position of a moving object varies with time. It captures the entire dynamics of the object's movement. Understanding this equation helps you predict where an object will be at any point in time and explain how its movement correlates to different forces acting upon it.For our particle's movement, the given equation is \( s = f(t) = 80t - 6t^2 \). This equation tells us:

• \( s \), the position of the particle, changes with time \( t \).
• The term \( 80t \) implies that without any opposing force, the particle moves linearly with time.
• The term \( -6t^2 \) indicates that as time progresses, some resistance or opposing force reduces the particle's forward motion.

This equation not only gives us a snapshot of the position at any time but, through differentiation, also offers insights into the velocity and speed, showing how comprehensive and useful equations of motion can be in analyzing real-world movement.