/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 A cannon on a railroad car is fa... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 25

A cannon on a railroad car is facing in a direction parallel to the tracks. It fires a \(98-\mathrm{kg}\) shell at a speed of \(105 \mathrm{m} / \mathrm{s}\) (relative to the ground) at an angle of \(60.0^{\circ}\) above the horizontal. If the cannon plus car have a mass of \(5.0 \times 10^{4} \mathrm{kg},\) what is the recoil speed of the car if it was at rest before the cannon was fired? [Hint: A component of a system's momentum along an axis is conserved if the net external force acting on the system has no component along that axis.]

Short Answer

Expert verified
Answer: The recoil speed of the car and the cannon is approximately -0.21 m/s, in the opposite direction of the horizontal component of the shell's velocity.

Step by step solution

01

Identify the initial momentum

Initially, before the cannon fires, the total momentum of the system (cannon+car+shell) is 0, as they are at rest.
02

Calculate the horizontal component of the shell's momentum

After the cannon fires, the shell has a momentum, which has a horizontal component. To calculate the horizontal component of the shell's momentum, we will use the formula: momentum_horizontal = mass_shell * velocity_horizontal Where mass_shell is the mass of the shell (98 kg), and velocity_horizontal is the horizontal component of the shell's velocity. To find the horizontal component of the velocity, we use the formula: velocity_horizontal = velocity * cos(angle) Where velocity is the total velocity of the shell (105 m/s), and angle is the angle it makes with the horizontal (60 degrees). Converting the angle to radians: angle_rad = (60 * pi) / 180 Calculating the horizontal velocity: velocity_horizontal = 105 * cos(angle_rad) Substituting the values: momentum_horizontal = 98 * velocity_horizontal
03

Calculate the recoil speed of the cannon+car

To find the recoil speed of the cannon+car, we will use the conservation of momentum in the horizontal direction. Since the initial momentum was 0, the final momentum of the cannon+car must be equal and opposite to the final momentum of the shell. Using the formula: momentum_cannon+car + momentum_shell = 0 momentum_cannon+car = -momentum_shell We then express the momentum of the cannon+car as the product of the mass and the recoil speed: mass_cannon+car * recoil_speed = -momentum_shell Solving for the recoil speed: recoil_speed = -momentum_shell / mass_cannon+car Substituting the values: recoil_speed = -momentum_horizontal / (5.0 * 10^4)
04

Calculate the numerical value for the recoil speed

Now, we can plug in the numerical values to find the recoil speed: angle_rad = (60 * pi) / 180 velocity_horizontal = 105 * cos(angle_rad) momentum_horizontal = 98 * velocity_horizontal recoil_speed = -momentum_horizontal / (5.0 * 10^4) Finally, calculating the recoil speed: recoil_speed 鈮� -0.21 m/s The negative sign indicates that the recoil speed of the cannon+car is in the opposite direction of the horizontal component of the shell's velocity.

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