Chapter 6: Dynamics I: Motion Along a Line

Seat belts and air bags save lives by reducing the forces exerted on the driver and passengers in an automobile collision. Cars are designed with a “crumple zone” in the front of the car. In the event of an impact, the passenger compartment decelerates over a distance of about 1 m as the front of the car crumples. An occupant restrained by seat belts and air bags decelerates with the car. By contrast, an unrestrained occupant keeps moving forward with no loss of speed (Newton’s first law!) until hitting the dashboard or windshield. These are unyielding surfaces, and the unfortunate occupant then decelerates over a distance of only about 5 mm.

a. A 60 kg person is in a head-on collision. The car’s speed at impact is 15 m/s. Estimate the net force on the person if he or she is wearing a seat belt and if the air bag deploys.

b. Estimate the net force that ultimately stops the person if he or she is not restrained by a seat belt or air bag.

Seat belts and air bags save lives by reducing the forces exerted on the driver and passengers in an automobile collision. Cars are designed with a “crumple zone” in the front of the car. In the event of an impact, the passenger compartment decelerates over a distance of about $1m$as the front of the car crumples. An occupant restrained by seat belts and air bags decelerates with the car. By contrast, an unrestrained occupant keeps moving forward with no loss of speed (Newton’s first law!) until hitting the dashboard or windshield. These are unyielding surfaces, and the unfortunate occupant then decelerates over a distance of only about $5mm$. a. A $60kg$person is in a head-on collision. The car’s speed at impact is $15m/s$. Estimate the net force on the person if he or she is wearing a seat belt and if the air bag deploys.

b. Estimate the net force that ultimately stops the person if he or she is not restrained by a seat belt or air bag

The piston of a machine exerts a constant force on a ball as it moves horizontally through a distance of $15cm$. You use a motion detector to measure the speed of five different balls as they come off the piston; the data are shown in the table. Use theory to find two quantities that, when graphed, should give a straight line. Then use the graph to find the size of the piston’s force.

Compressed air is used to fire a 50 g ball vertically upward from a 1.0 m-tall tube. The air exerts an upward force of 2N on the ball as long as it is in the tube. How high does the ball go above the top of the tube?

a. A rocket of mass $m$is launched straight up with thrust ${\overrightarrow{F}}_{thrust}$. Find an expression for the rocket’s speed at height $h$if air resistance is neglected.

b. The motor of a $350$g model rocket generates$9.5$ N thrust. If air resistance can be neglected, what will be the rocket’s speed as it reaches a height of $85$ m?

A rifle with a barrel length of $60$cm fires a $10$g bullet with a horizontal speed of $400$m/s. The bullet strikes a block of wood and penetrates to a depth of $12$cm.

a. What resistive force (assumed to be constant) does the wood exert on the bullet?

b. How long does it take the bullet to come to rest?

A truck with a heavy load has a total mass of $7500$kg. It is climbing a $15°$incline at a steady $15$m/s when, unfortunately, the poorly secured load falls off! Immediately after losing the load, the truck begins to accelerate at $1.5$ m/s². What was the mass of the load? Ignore rolling friction.

An object of mass m is at rest at the top of a smooth slope of height $h$and length$L$. The coefficient of kinetic friction between the object and the surface, ${\mu }_k$, is small enough that the object will slide down the slope after being given a very small push to get it started. Find an expression for the object’s speed at the bottom of the slope.

Sam, whose mass is $75$kg, takes off across level snow on his jet-powered skis. The skis have a thrust of $200$N and a coefficient of kinetic friction on snow of $0.10$. Unfortunately, the skis run out of fuel after only$10$ s.

a. What is Sam’s top speed?

b. How far has Sam traveled when he finally coasts to a stop?

An elevator, hanging from a single cable, moves upward at constant speed. Friction and air resistance are negligible. Is the tension in the cable greater than, less than, or equal to the gravitational force on the elevator? Explain. Include a free-body diagram as part of your explanation.