Problem 37 - Calculus Early Transcendentals

Question

\(37-38\) 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 \(=5\) $$f(t)=100+50 t-4.9 t^{2}$$

Step-by-Step Solution

Answer

Velocity is 1 m/s, and speed is 1 m/s.

Step 1: Differentiate to Find Velocity

The velocity of the particle is given by the derivative of the position function with respect to time. Calculate the derivative of \(f(t) = 100 + 50t - 4.9t^2\) to get \(v(t)\). The first derivative is:\[v(t) = \frac{d}{dt}(100 + 50t - 4.9t^2) = 0 + 50 - 9.8t\]So, \(v(t) = 50 - 9.8t\).

Step 2: Evaluate Velocity at t = 5

Substitute \(t = 5\) into the velocity function \(v(t) = 50 - 9.8t\) to find the velocity at this specific time:\[v(5) = 50 - 9.8(5) = 50 - 49 = 1 \, \text{m/s}\]Thus, the velocity at \(t = 5\) is \(1 \, \text{m/s}\).

Step 3: Determine Speed at t = 5

Speed is the absolute value of velocity. Since the velocity at \(t = 5\) is positive, the speed is the same as the velocity.\[\text{Speed} = |v(5)| = |1| = 1 \, \text{m/s}\]

Key Concepts

Understanding Particle Motion

When discussing particle motion, we're focusing on how a particle travels over a specific pathway concerning time. In many calculus problems, especially those involving motion, we define the position of a particle with a function of time. This function, often represented as \(s = f(t)\), indicates the position in meters at any given time \(t\) in seconds. Tracking the movement of particles is crucial in physics and engineering to understand how objects move and predict future positions.

• If the function is quadratic, like \(f(t) = 100 + 50t - 4.9t^2\), the path of the particle is a curve.
• The motion isn't uniform as the velocity changes with time.

Considering particle motion helps us model real-world phenomena, such as how long it takes for an object to fall to the ground.

Unraveling Velocity Calculation

Calculating velocity is a fundamental step in understanding particle motion. Velocity tells us the rate of change of the position over time.
To find this, we look for the derivative of the position function, \(f(t)\). This derivative, denoted as \(v(t)\), reveals how the position \(s\) changes as time progresses.
In our example, by differentiating \(f(t) = 100 + 50t - 4.9t^2\), we get \(v(t) = 50 - 9.8t\). This represents the velocity function, indicating that velocity decreases as time progresses.

• When \(t = 5\), substituting into \(v(t)\) results in a velocity of 1 m/s.
• This means at that moment, the particle is moving forward at 1 meter per second.

Understanding how to calculate velocity provides the foundation for analyzing how rapid or slow a particle moves over time.

Clarifying Speed Determination

Once you determine velocity, finding the speed is straightforward. Speed is the magnitude of velocity, representing how fast an object is moving regardless of direction.

• Mathematically, it is the absolute value of velocity, \(|v(t)|\).
• For our particle at \(t = 5\), with a velocity of 1 m/s, the speed is also 1 m/s because the velocity is positive.

Speed is always a non-negative quantity, as it doesn't consider direction, only magnitude. By understanding speed, we focus solely on how fast an object is moving, which is pertinent in contexts like determining travel time or ensuring objects stay within speed limits.
In our example, with a speed of 1 m/s at \(t = 5\), it merely conveys the intensity of motion.