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Chapter 2: Problem 70

A castle's defenders throw rocks down on their attackers from a 15-m-high wall, with initial speed 10 m/s. How much faster are the rocks moving when they hit the ground than if they were simply dropped?

Short Answer

Expert verified
Calculate the difference in final speed \( v_{f_{throw}} - v_{f_{drop}} \) for the short answer.

Step by step solution

01

Calculate the falling time

First calculate the time taken for the rocks to fall. This can be done using the equation \( h = gt^2/2 \) for free fall (where gravity \(g = 9.8 m/s^2\), and the height \(h = 15m\)). Solving this equation for \(t\), the time taken for the rocks to fall, gives \( t = sqrt(2h/g)\).
02

Calculate final speed when dropped

To find out how fast the rocks are moving when they hit the ground when dropped, substitute \( v_i = 0 \) (initial speed is zero since the rock is dropped) and the calculated time value into the equation for final velocity \( v_f = v_i + gt \). Calculate the final speed \( v_{f_{drop}} \).
03

Calculate final speed when thrown

Now find out how fast the rocks are moving when they hit the ground when thrown down. Substitute \( v_i = 10 m/s \) (initial speed when the rock is thrown down) and the calculated time value into the equation for final velocity \( v_f = v_i + gt \). Calculate the final speed \( v_{f_{throw}} \).
04

Calculate the difference in final speed

Finally, subtract the final speed when dropped from the final speed when thrown to find out how much faster the rocks are moving when they hit the ground when thrown than when simply dropped. This is given by \( v_{f_{throw}} - v_{f_{drop}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Free Fall Motion
Free fall motion describes the movement of an object under the influence of gravity alone. It's a fascinating concept in physics that demonstrates how an object accelerates when it's dropped or thrown vertically downward from a certain height thanks to Earth's gravitational pull, which is approximately measured as 9.8 m/s2.

During free fall, air resistance is often disregarded to simplify calculations. This assumption makes it possible to predict an object's behavior under gravity's unopposed force. The object accelerates downwards at a constant rate, meaning its velocity increases the longer it falls, which directly affects the final velocity 鈥� the speed at which it will hit the ground.

In the case of the castle defenders, rocks thrown down verses those simply dropped represent two different free fall scenarios. Although both are subject to gravity, their initial velocities differ. This discrepancy is key when determining the impact velocity of each rock, as the one thrown down initially will gain additional speed due to both the initial throw and gravitational acceleration.
Final Velocity Calculation
The final velocity calculation is crucial in understanding how an object's speed evolves during free fall. It takes into account initial velocity, acceleration due to gravity, and the time for which the object has been falling.

The formula for calculating final velocity is vf = vi + gt, where vf represents the final velocity, vi is the initial velocity (which is zero for an object simply dropped), g stands for the acceleration due to gravity, and t is the time elapsed. This equation shows that the final velocity is a result of both the initial push (if any) and the acceleration experienced over time.

For the defenders' rocks, the initial speeds are different: one rock starts at rest when dropped, while the other has a preliminary speed from being thrown. This causes the thrown rock to have a greater final velocity, translating to a greater impact force upon reaching the ground.
Kinematic Equations
Kinematic equations allow us to calculate various aspects of an object's motion, like its displacement, velocity, and time taken to travel a certain distance. These equations form the backbone of classical mechanics and are widely applicable in solving physics problems that involve motion with constant acceleration.

One such kinematic equation, h = vit + (1/2)gt2, relates the height from which an object falls with the time it takes to reach the ground. This equation, along with the final velocity formula, was used in the textbook solution to determine the time it took for the rocks to reach the ground and their respective final velocities.

By understanding and applying kinematic equations, one can derive not only the final velocities but also the differential speed鈥攈ow much faster one rock travels compared to the other when they hit the ground. In summary, these equations are a powerful set of tools that enable students to decode motion scenarios in a systematic and predictive manner.

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Most popular questions from this chapter

An X-ray tube gives electrons constant acceleration over a distance of \(15 \mathrm{cm} .\) If their final speed is \(1.2 \times 10^{7} \mathrm{m} / \mathrm{s},\) what are (a) the electrons' acceleration and (b) the time they spend accelerating?

Starting from home, you bicycle 24 km north in 2.5 h and then turn around and pedal straight home in \(1.5 \mathrm{h}\). What are your (a) displacement at the end of the first \(2.5 \mathrm{h},\) (b) average velocity over the first \(2.5 \mathrm{h},\) (c) average velocity for the homeward leg of the trip, (d) displacement for the entire trip, and (e) average velocity for the entire trip?

An object's acceleration is given by the expression \(a(t)=-a_{0} \cos \omega t,\) where \(a_{0}\) and \(\omega\) are positive constants. Find expressions for the object's (a) velocity and (b) position as functions of time. Assume that at time \(t=0\) it starts from rest at its greatest positive displacement from the origin. (c) Determine the magnitudes of the object's maximum velocity and maximum displacement from the origin.

The position of a car in a drag race is measured each second, and A the results are tabulated below. $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Time } t(\mathrm{s}) & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Position } x(\mathrm{m}) & 0 & 1.7 & 6.2 & 17 & 24 & 40 \\ \hline \end{array}$$ Assuming the acceleration is approximately constant, plot position versus a quantity that should make the graph a straight line. Fit a line to the data, and from it determine the approximate acceleration.

You're a consultant on a movie set, and the producer wants a car to drop so that it crosses the camera's field of view in time At. The field of view has height \(h .\) Derive an expression for the height above the top of the field of view from which the car should be released.
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