Problem 49
Question
When an object is dropped, the distance it falls in \(t\) seconds, assuming negligible air resistance, is given by $$ s(t)=16 t^{2} $$ where \(s(t)\) is in feet. Suppose a medic's reflex hammer falls from a hovering helicopter. Find (a) how far the hammer falls in \(3 \mathrm{sec},(\mathrm{b})\) how fast the hammer is traveling 3 sec after being dropped, and (c) the hammer's acceleration after it has been falling for 3 sec.
Step-by-Step Solution
Verified Answer
In 3 seconds, the hammer falls 144 feet, travels at 96 ft/s, and its acceleration is 32 ft/s².
1Step 1: Calculate the Distance Fallen in 3 Seconds
To find the distance the hammer falls in 3 seconds, we substitute \( t = 3 \) into the given formula \( s(t) = 16t^2 \). Thus, \( s(3) = 16 \times 3^2 = 16 \times 9 = 144 \). The hammer falls 144 feet in 3 seconds.
2Step 2: Calculate the Speed After 3 Seconds
The speed of the hammer, or its velocity, is the derivative of the distance function with respect to time, which is \( v(t) = \frac{ds}{dt} = \frac{d}{dt}[16t^2] = 32t \). To find the speed at 3 seconds, substitute \( t = 3 \) into the derivative: \( v(3) = 32 \times 3 = 96 \). So, the speed of the hammer after 3 seconds is 96 feet per second.
3Step 3: Determine the Acceleration After 3 Seconds
The acceleration is the derivative of the velocity function, which is a constant: \( a(t) = \frac{dv}{dt} = \frac{d}{dt}[32t] = 32 \). Thus, the hammer's acceleration is 32 feet per second squared at any time after being released, including at 3 seconds.
Key Concepts
DerivativesPhysics applicationsKinematics
Derivatives
In calculus, a derivative represents the rate of change of a function with respect to a variable. When dealing with motion, like in this exercise, the derivative tells us how quantities such as distance and velocity change over time.
- The **first derivative** of the distance function gives us the velocity function. This tells us how fast the distance is changing with time. For the function \( s(t) = 16t^2 \), the first derivative is \( v(t) = \frac{ds}{dt} = 32t \). This means that the velocity is directly proportional to time, indicating a constant acceleration.
- The **second derivative** of the distance function, or the first derivative of the velocity function, provides us with the acceleration. For our initial velocity function \( v(t) = 32t \), the derivative is \( a(t) = \frac{dv}{dt} = 32 \). Thus, the acceleration is constant at 32 feet per second squared. Understanding derivatives is crucial for solving problems in physics, engineering, and other fields where change is a key factor.
- The **first derivative** of the distance function gives us the velocity function. This tells us how fast the distance is changing with time. For the function \( s(t) = 16t^2 \), the first derivative is \( v(t) = \frac{ds}{dt} = 32t \). This means that the velocity is directly proportional to time, indicating a constant acceleration.
- The **second derivative** of the distance function, or the first derivative of the velocity function, provides us with the acceleration. For our initial velocity function \( v(t) = 32t \), the derivative is \( a(t) = \frac{dv}{dt} = 32 \). Thus, the acceleration is constant at 32 feet per second squared. Understanding derivatives is crucial for solving problems in physics, engineering, and other fields where change is a key factor.
Physics applications
Calculus helps us solve numerous real-world problems, particularly in physics. The exercise involving the falling reflex hammer is an example of how kinematic equations in physics are underpinned by calculus.
- **Finding distance**: The distance fallen by an object under gravity (with negligible air resistance) can be calculated using a quadratic function such as \( s(t) = 16t^2 \). The coefficient 16 relates to half the acceleration due to gravity, commonly approximated as 32 feet per second squared.
- **Determining speed and acceleration**: By finding the derivatives, we smoothly transition from distance to speed, and then to acceleration. In the context of physics, the first derivative of the distance function (velocity) shows how fast an object is moving, while the second derivative (acceleration) reveals how the speed changes over time. These calculations can be vital for designing safe transportation systems and understanding how objects behave in free fall.
- **Finding distance**: The distance fallen by an object under gravity (with negligible air resistance) can be calculated using a quadratic function such as \( s(t) = 16t^2 \). The coefficient 16 relates to half the acceleration due to gravity, commonly approximated as 32 feet per second squared.
- **Determining speed and acceleration**: By finding the derivatives, we smoothly transition from distance to speed, and then to acceleration. In the context of physics, the first derivative of the distance function (velocity) shows how fast an object is moving, while the second derivative (acceleration) reveals how the speed changes over time. These calculations can be vital for designing safe transportation systems and understanding how objects behave in free fall.
Kinematics
Kinematics is a branch of mechanics that describes the motion of objects without considering the causes of motion. In our exercise, the motion of a reflex hammer is analyzed using kinematic equations.
- **Distance computation**: The equation \( s(t) = 16t^2 \) is an example of a kinematic equation used to calculate how far an object has traveled over time. It considers constant acceleration, which is typical when gravity is involved without any other resistance.
- **Velocity analysis**: Understanding velocity as the rate of change of distance informs us about the uniformity and direction of an object’s motion. After 3 seconds, the hammer reaches a velocity of 96 feet per second, showing a steady increase due to gravitational pull.
- **Uniform acceleration**: In this scenario, the constant acceleration value of 32 feet per second squared simplifies calculations. Kinematic equations that assume constant acceleration help predict and describe motion effectively, making them crucial tools in physics and engineering.
- **Distance computation**: The equation \( s(t) = 16t^2 \) is an example of a kinematic equation used to calculate how far an object has traveled over time. It considers constant acceleration, which is typical when gravity is involved without any other resistance.
- **Velocity analysis**: Understanding velocity as the rate of change of distance informs us about the uniformity and direction of an object’s motion. After 3 seconds, the hammer reaches a velocity of 96 feet per second, showing a steady increase due to gravitational pull.
- **Uniform acceleration**: In this scenario, the constant acceleration value of 32 feet per second squared simplifies calculations. Kinematic equations that assume constant acceleration help predict and describe motion effectively, making them crucial tools in physics and engineering.
Other exercises in this chapter
Problem 49
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Consider the function \(f\) given by $$f(x)=\frac{x^{2}-9}{x+3}$$ a) For what \(x\) -value(s) is this function not differentiable? b) Find \(f^{\prime}(4)\).
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Find \(\frac{d y}{d u}, \frac{d u}{d x},\) and \(\frac{d y}{d x}\) $$ y=(u+1)(u-1) \text { and } u=x^{3}+1 $$
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