Newtonian Mechanics - AP Physics 1

The period describes how long it takes the satellite to make one full orbit. If you go back to the definition of velocity, , we can apply that to our new circular orbit, in which the distance is equal to the circumference of the circle and the time is equal to the period: . The circumference divided by the period will give us the average velocity.

The problem gives us the radius, but we need to find the tangential velocity. We can do this by first solving for the centripetal acceleration from the centripetal force.

Recognize that the force due to gravity of the Earth on the satellite is the same as the centripetal force acting on the satellite. That means .

Solve for for the satellite. To do this, use the law of universal gravitation.

Remember that is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

Now that we know the force, we can find the acceleration. Remember that centripetal force is . Set our two forces equal and solve for the centripetal acceleration.

Now we can find the tangential velocity, using the equation for centripetal acceleration. Again, remember that the radius is equal to the sum of the radius of the Earth and the height of the satellite!

We now have a value for the tangential velocity, which we can use in the equation for velocity from the beginning to find the period.

We can use the conservation of momentum to solve. Since the satellites stick together, there is only one final velocity term.

We know the masses for both satellites are equal, and the second satellite is initially stationary.

Now we need to find the velocity of the first satellite. Since the satellite is in orbit (circular motion), we need to find the tangential velocity. We can do this by finding the centripetal acceleration from the centripetal force.

Recognize that the force due to gravity of the Earth on the satellite is the same as the centripetal force acting on the satellite. That means .

Solve for for the satellite. To do this, use the law of universal gravitation.

Remember that is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

Now that we know the force, we can find the acceleration. Remember that centripetal force is . Set our two forces equal and solve for the centripetal acceleration.

Now we can find the tangential velocity, using the equation for centripetal acceleration. Again, remember that the radius is equal to the sum of the radius of the Earth and the height of the satellite!

This value is the tangential velocity, or the initial velocity of the first satellite. We can plug this into the equation for conversation of momentum to solve for the final velocity of the two satellites.

Since the collision is completely inelastic, momentum is conserved but energy is not. Furthermore, the two cars stick to each other and travel as one. The equation for conservation of momentum is as follows:

There are two inital masses with different velocities and one final mass with a single velocity. Therefore, we can write:

Rearranging for final velocity, we get:

At this point, we can denote which direction is positive and which is negative. Since the car traveling west has more momentum, we will consider west to be positive. Substituting our values into the equation, we get:

Since this value is positive, the final answer is West.

The correct answer is "Newton's First Law: every object continues in its state of rest or of uniform velocity in a straight line as long as no net force acts on it." On a slippery surface, we can assume there is no friction. If there is no friction working against the box's movement, the box will continue to move at its initial velocity forever. If the box were on a rough surface, friction from the surface would act against the box, causing the box's velocity to slow and eventually stop (if the force acting on the box is not greater than that of the frictional force).

According to Newton's first law, an object will remain at rest until a force is applied. If the floor of the boxcar is perfectly motionless, than the object will remain in the same spot in relation to the earth. This will make it appear to move to the back of the boxcar.

Force is defined as

Since the velocity is constant, the acceleration is zero. Therefore the force required to keep this object in motion is zero.

This is also stated in Newton's First Law: "An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force."

According to Newton's First Law of Motion, an object that is in motion will stay in motion unless acted on by another force. When the van slows down, the ball will want to continue moving forward, and the friction between it and the floor of the van is not strong enough to keep the ball back.

The car and the truck experience equal and opposite forces, but since the car has a smaller mass it will experience greater acceleration than the truck according to the equation F = ma.

A greater mass will decrease the acceleration.

We need to use the equation for impulse to solve this problem. In fact, the time given is completely irrevelevant:

Plugging in our values:

This is less than the threshold, so no, nothing becomes structurally compromised

Since the collision is completely elastic, we know that both momentum and kinetic energy are conserved. We can write the following equations (initial momentum and energy of the second ball are neglected since it is not moving:

Rearrange the first equation for and the second for .

We can rewrite the second equation as:

Substitute our equation for into the second equation:

Rearranging, we get:

Plug in our values for the initial and final velocities:

To solve for mass, we'll use our earlier expression for :

Impulse can be written as either of two popular expressions:

From the problem statement, we can determine the velocity of the marble as it hits the floor, allowing us to use the latter expression. To determining the velocity of the marble, we can use the equation for conservation of energy:

Assuming the final height is zero, we can eliminate initial kinetic energy and final potential energy. Therefore, we can write:

Canceling out mass and rearranging for final velocity, we get:

We know these variables, allowing us to solve for the velocity:

Plugging this value into the expression for impulse, we get:

Momentum is always conserved. Equation for conservation of momentum:

There is only one velocity on the right since the two astronauts grab onto each other, thus they move together at the same velocity. Solve.

Impulse is just the change in momentum. To find the velocity when the ball hits the ground, we need to use kinematics. We know the height the ball is dropped, the acceleration, and the initial velocity, so we can use the equation . The initial velocity is , , and , so the equation becomes

When the ball bounces back up it reaches a height of . In order to find the velocity immediately after it hits the ground, we can use the same equation with . This will lead it a velocity of

Assuming up is positive, the magnitude of the impulse is just

To calculate the momentum of the block, we first need to know the velocity of the block. This can be found using the equation for the conservation of momentum:

If we assume that the final height is zero, we can eliminate initial kinetic energy and final potential energy, getting:

Substituting expressions for each term, we get:

Cancel out mass and rearrange to solve for velocity:

We can use the horizontal distance traveled and the angle of the slope to determine the initial height:

Now that we have the initial height, we can solve for final velocity:

Finally, we can now use the equation for momentum to solve the problem:

The equation for kinetic energy is:

, where is the velocity of the object. Therefore, if we halve the velocity, we can substitute that into our equation, and see what will change in the kinetic energy equation:

Thus, the kinetic energy is quartered.

To find the acceleration of the box traveling down the incline, the mass is not needed. Using the incline of the plane as the x-direction, we can see that there is no movement in the y-direction; therefore, we can use Newton's second, F = ma, in the x-direction.

There is only one force in the x-direction (gravity), however gravity is not just equal to “mg” in this case. Since the box is on an incline, the gravitational force will be equal to mgsin(30o). Substituting force into F =ma we find that mgsin(30o) = ma. We can now cancel out masses and solve for acceleration.

Since the spring is Hookean, the relationship between the force and displacement from equilibrium of the mass can be expressed by Hooke's Law:

Since this equation is linear, the force and displacement are directly proportional. Thus, when the displacement doubles, the force doubles.

We need our kinematic equation that relates displacement, initial velocity, and acceleration:

We can neglect gravity in this problem since the statement says that the missile accelerates in a straight line at a constant rate.

Plugging in our values, we get:

There are two ways to solve this problem. The first and much easier way is to use the range equation. The second is using your kinematics equations.

Method 1: Range Equation

The range equation is the following:

We know everything ecvept for the initial velocity. However, we can calculate it knowing that the cannon transfers 500 J of energy to the ball. Therefore:

Rearranging for velocity:

Now we can plug everything into the range equation:

Method 2: Kinematics Equations

As in the first method, we can calculate the initial velocity of the ball:

Rearranging for velocity:

We can then split this into it's components:

We can use the y-component to calculate how long the ball is in the air. We can do one of two things:

1. Calculate how long the ball takes to reach it's peak and then multiply by 2
2. Calculate how lon gthe ball takes to get a velocity of -5

We'll go with method 2:

Rearranging for t:

Then we can multiply this time by the horizontal velocity (which stays constant because we are neglecting air resistance).

We know that the plane needs to drop 3,000 meters in 10 minutes. Therefore, we can calculate the vertical velocity that the plane needs to be traveling at:

Knowing the plane's velocity, we can caluclate what angle at which it needs to decend to achieve this vertical velocity:

Rearranging for theta:

Plugging in our values: