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Chapter 12: Problem 8

A projectile with a mass of \(2.40 \mathrm{~kg}\) is fired from a cliff \(125 \mathrm{~m}\) high with an initial velocity of \(150 \mathrm{~m} / \mathrm{s}\), directed \(41.0^{\circ}\) above the horizontal. What are \((a)\) the kinetic energy of the projectile just after firing and \((b)\) its potential energy? \((c)\) Find the speed of the projectile just before it strikes the ground. Which answers depend on the mass of the projectile? Ignore air drag.

Short Answer

Expert verified
The kinetic energy just after firing is 27000 J. The potential energy is 29412 J. The speed of the projectile just before it strikes the ground is 151.4m/s. Both kinetic and potential energies depend on the mass of the projectile while the final speed does not.

Step by step solution

01

Calculate the kinetic energy just after firing

To find out the kinetic energy right after the firing, the following formula can be used:\n Kinetic energy = 0.5 * mass * (velocity)^2. Here, mass equals 2.4kg and velocity equals 150 m/s. Substituting these values into the formula yields:\n Kinetic energy = 0.5 * 2.4kg * (150m/s)^2.
02

Calculate the potential energy

To calculate potential energy, the formula potential energy = mass * g * height is used, where g is gravity (approx. 9.81 m/s^2) and height is the distance above the ground (125m). So, the potential energy = 2.4kg * 9.81m/s^2 * 125m.
03

Calculate the speed of the projectile just before it hits the ground

The initial vertical and horizontal velocities can be calculated using trigonometry, considering the projectile's velocity and the launch angle. The formula for the final speed of the projectile when it is about to strike the ground is given by the Pythagorean theorem (horizontal_vel^2 + vertical_vel^2 = final_vel^2), where horizontal_vel is the horizontal_velocity * time and vertical_vel is the square root of 2*g*h. Substituting the calculated values of horizontal and vertical velocities into this formula will give the final velocity.
04

Evaluate which answers depend on the mass of the projectile

In this problem, both the kinetic and potential energies are directly proportional to the mass of the projectile. Hence, both of these values will change with a change in mass. However, the final velocity of the projectile does not depend on its mass.

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Most popular questions from this chapter

A pendulum is made by tying a \(1.33-\mathrm{kg}\) stone to a string \(3.82 \mathrm{~m}\) long. The stone is projected perpendicular to the string, away from the ground, with the string at an angle of \(58.0^{\circ}\) with the vertical. It is observed to have a speed of \(8.12 \mathrm{~m} / \mathrm{s}\) when it passes its lowest point. (a) What was the speed of the stone when projected? (b) What is the largest angle with the vertical that the string will reach during the stone's motion? (c) Using the lowest point of the swing as the zero of gravitational potential energy, calculate the total mechanical energy of the system.

A particle moves along the \(x\) axis under the influence of a conservative force that is described by $$\overrightarrow{\mathbf{F}}=-\alpha x e^{-\beta x^{2}} \hat{\mathbf{i}}$$ where \(\alpha\) and \(\beta\) are constants. Find the potential energy function \(U(x)\).

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An object falls from rest from a height \(h\). Determine the kinetic energy and the potential energy of the object as a function \((a)\) of time and \((b)\) of height. Graph the expressions and show that their sum - the total mechanical energy - is constant in each case.
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