Problem 17
Question
The height \(s\) (in feet) at time \(t\) (in seconds) of a silver dollar dropped from the top of the Washington Monument is given by \(s=-16 t^{2}+555\) (a) Find the average velocity on the interval \([2,3]\). (b) Find the instantaneous velocities when \(t=2\) and when \(t=3\) (c) How long will it take the dollar to hit the ground? (d) Find the velocity of the dollar when it hits the ground.
Step-by-Step Solution
Verified Answer
The velocities: (a) Average velocity on the interval [2,3] can be calculated by substituting the given times into the given equation and calculating the resultant value. (b) The instantaneous velocities at \(t=2\) and \(t=3\) can be determined using the derivative of the equation, substituting the given times, and calculating the resultant values. (c) The time when the dollar hits the ground can be calculated by setting the equation to 0 and solving for \(t\). (d) The velocity when the dollar hits the ground can be found by substituting the time at which the dollar hits the ground into the equation for the derivative.
1Step 1: Calculate the average velocity on the interval [2,3]
The formula for average velocity is \(\frac{\Delta s}{\Delta t}\) which in this scenario will be \(\frac{s(3) - s(2)}{3 - 2}\) by substituting \(s=-16 t^{2}+555\)
2Step 2: Find the instantaneous velocities when t=2 and when t=3
The formula for instantaneous velocity at time t is the derivative of the s(t) equation, \(s'(t)\). So calculate the derivative of the given equation, then substitute the values for t=2 and t=3.
3Step 3: Find when the dollar hits the ground
The height s is zero when the dollar hits the ground, so set the equation \(s= -16t^2 + 555\) to zero and solve for \(t\).
4Step 4: Find the velocity of the dollar when it hits the ground
Substitute the \(t\) value found from Step 3 into the equation for the instantaneous velocity, \(s'(t)\), found in Step 2.
Key Concepts
Understanding Instantaneous VelocityGrasping Average VelocityThe Role of Derivative in Kinematics
Understanding Instantaneous Velocity
Instantaneous velocity is all about the speed and direction of an object at a specific moment in time. It's like taking a snapshot of the object’s speed at an exact second. To calculate this in calculus, we use the derivative of the position function. This gives us the exact rate of change of position at that point. For example, in our exercise, the position function is given by \(s = -16t^2 + 555\).
To find the instantaneous velocity, we first take the derivative of this function, which gives us the velocity function \(s'(t)\). Plugging a specific time into \(s'(t)\) tells us the velocity at that exact moment.
So, if you want to know how fast the silver dollar is falling at \(t = 2\) or \(t = 3\), just substitute these values into the velocity function. This derivative provides the mathematical way to see the speed at which an object is moving, making instant velocity very real-world applicable!
To find the instantaneous velocity, we first take the derivative of this function, which gives us the velocity function \(s'(t)\). Plugging a specific time into \(s'(t)\) tells us the velocity at that exact moment.
So, if you want to know how fast the silver dollar is falling at \(t = 2\) or \(t = 3\), just substitute these values into the velocity function. This derivative provides the mathematical way to see the speed at which an object is moving, making instant velocity very real-world applicable!
Grasping Average Velocity
Average velocity is like looking at a longer stretch of an object's journey. It tells us the overall speed of the object over a period of time by considering how far it traveled and the time it took.
Think of it as the total change in position divided by the total time. Mathematically, it's expressed as \(\frac{\Delta s}{\Delta t}\). In our example, this becomes \(\frac{s(3) - s(2)}{3 - 2}\).
This formula gives a single number that represents the entire interval. It smooths out any speeding up or slowing down to give a big picture view of the object's motion. This concept is a handy way to get a sense of overall movement without diving into moment-by-moment details.
Think of it as the total change in position divided by the total time. Mathematically, it's expressed as \(\frac{\Delta s}{\Delta t}\). In our example, this becomes \(\frac{s(3) - s(2)}{3 - 2}\).
This formula gives a single number that represents the entire interval. It smooths out any speeding up or slowing down to give a big picture view of the object's motion. This concept is a handy way to get a sense of overall movement without diving into moment-by-moment details.
The Role of Derivative in Kinematics
Derivatives are fundamental in describing how things change, making them crucial in understanding motion. When we talk about a function that shows how position changes over time, the derivative of this function shows how velocity changes.
In the context of kinematics, the derivative essentially translates a position function into a velocity function. In our problem, taking the derivative of \(s = -16t^2 + 555\) gives us \(s'(t)\), which represents the velocity.
This process captures the idea of immediate speed. Derivatives help us move from the broad strokes of motion to the particular speed and acceleration, providing a powerful mathematical tool to understand dynamic systems in physics. Whether we need to know the instant speed at a specific time or how quickly the speed itself changes, derivatives offer that insight.
In the context of kinematics, the derivative essentially translates a position function into a velocity function. In our problem, taking the derivative of \(s = -16t^2 + 555\) gives us \(s'(t)\), which represents the velocity.
This process captures the idea of immediate speed. Derivatives help us move from the broad strokes of motion to the particular speed and acceleration, providing a powerful mathematical tool to understand dynamic systems in physics. Whether we need to know the instant speed at a specific time or how quickly the speed itself changes, derivatives offer that insight.
Other exercises in this chapter
Problem 16
Find the limit of (a) \(\sqrt{f(x)}\), (b) \([3 f(x)]\), and (c) \([f(x)]^{2}\), as \(x\) approaches \(c\). $$ \lim _{x \rightarrow c} f(x)=9 $$
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Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d)
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Find the derivative of the function. $$ y=4 t^{4 / 3} $$
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Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, id
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