Newtonian Mechanics - AP Physics 1

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:

We can use our kinematics equations to determine the height of the rocket the moment it loses thrust:

If we clarify the initial state being the moment the rocket loses thrust and the final state being the point of its maximum height, we can use the expression for conservation of energy to calculate the final height:

Removing final kinetic energy (which will be zero) and substituting in our expressions for each variable:

Cancel out mass and rearrange for final height:

From the problem statement, we can determine the initial velocity:

Now that we have values for all of our variables, we can solve:

Displacement has magnitude and direction. It is found by subtracting the position at time zero from the position at time ten seconds.

There are two forces in play in this scenario: friction and gravity. We can use Newton's second law to develop an expression for the acceleration of the block:

Substituting in expressions for each force, we get:

Canceling out mass and rearranging for acceleration, we get:

We know all of our values, allowing us to solve for the acceleration:

We can use the equation for conservation of energy to solve this problem:

If we assume the initial state is when the slingshot is fully loaded and the final state is at the ball's maxmium height, we can completely eliminate kinetic energy to get:

Substituting in our value for initital potential energy and an expression for final potential energy, we get:

Rearranging for final height we get:

If an object is moving to the right, then its displacement is positive. If its displacement is positive then its velocity must be positive as well. The object is speeding up. This implies that the magnitude of its velocity is increasing. If this is true we have that the velocity of the object, , after some time, , has passed must be a larger positive velocity than say, , the object's velocity before that time passed. Therefore, the object's average acceleration is given by:

since

So we have that the object's acceleration is positive. If the object's acceleration is positive, then there must be a positive net force causing the acceleration, due to Newton's first law: . Therefore, it is false that the object experiences a net force of 0N.

If the object is moving to the left, then its displacement must be negative since its final position will be further to the left in than its initial position. So in an imaginary number line, after some time passes, the object's final position is less than the object's initial position. This can be shown mathematically:

, since

Since the object's displacement is negative, then so is its velocity. This can easily be seen from the equation for average velocity:

since . At the end of its trajectory, the object stops, so overall the object's velocity is less than or equal to .

The object eventually stops. This means that the magnitude of the object's velocity is decreasing. In other words, the absolute value of the object's velocity is becoming smaller and smaller until it stops. So, for example, if at some moment the object's velocity is , after some time its velocity will be . Remember that the negative sign indicates only the direction of the vector (the object is moving to the left, hence its velocity is negative). But in absolute terms, the object is moving slower and slower (to the left) until it stops. Finally, we can see that the acceleration must be positive in order to slow down an object moving to the left. Going of our example, let's say the object is moving at and after some time it slows down to (again the negative sign just indicates that the object moves to the left). So, if look at the average acceleration we have:

Thus the acceleration must be positive.

It does not matter whether the collision is elastic or inelastic (although it would be best to assume that it's inelastic). Momentum is conserved in either type of collision, and is the only value needed for our calculation. Since they come to a standstill, their momentums at the moment of collision are equal and opposite:

Rearrange to solve for :

Plug in the given values from the question and solve:

By definition, acceleration is the rate of change of velocity with respect to time. The average acceleration of an object is given by:

Therefore, if velocity is constant it must be the case that acceleration is 0, since the initial and final velocities of the object are equal. Similarly, if acceleration is non-zero, then it must be the case that the initial and final velocities of the object are different (so velocity is changing). Now, when an object's acceleration and initial-velocity have the same sign (i.e. the same direction) then the object will speed up. In other words, the absolute value of its velocity (the magnitude) increases. This is easy to see if we solve for the final velocity in our equation:

If both and have the same sign, then the absolute value of will be greater as time passes, meaning the object is speeding up. (Try plugging in some numbers). Similarly, when and have opposite signs, the magnitude of the object's velocity must start to decrease. After enough time it might even come to a stop, and if it continues to experience the acceleration it will change direction and start gaining velocity on the opposite direction. Regardless, it must first slow down (think about when you throw an object on the air).

Example:

Example:

Remember that the positive and negative signs indicate the direction of the vectors. So when it comes to slowing down or speeding up, we want to look at the magnitude of the velocity. Therefore, the only statement that is false is the one that says that if an object has both negative initial velocity and acceleration it will slow down. The object will speed up and continue to move to the left.

The slope of a distance vs. time plot represents the object's velocity. Therefore, in both the first and second graphs, the velocity is constant and zero, respectively. An object with constant (or no) velocity is experiencing, by definition, no net force. Oppositely, in the third graph the slope is increasing, meaning the object is accelerating, meaning it must be subject to some force.

The net force experienced by m1 can be characterized as:

where T is the tension force along the string and mg is gravity.

because is also equivalent to , we can express this equation as

.

The net force experienced by m2 can similarly be expressed as:

and so

From this point the problem is a system of equations.

If we solve for T in the first equation, we end up with .

Then we can plug into the second equation like so:

then:

finally we can isolate a:

when we plug in the masses given by the problem, we get:

The basic kinematics equation for motion under constant acceleration is:

We do not have the acceleration, but we can find it:

Apply this to the first equation and solve.

The equation for speed is:

The runner has gone around the track 1.5 times, thus his total distance traveled is:

We can find the elapsed time from the walker: Rearrange the speed equation to find time:

Now we have all the information we need to find the runner's speed:

To find how fast the driver was going before applying the brakes, we use the following kinematics formula since acceleration is constant:

Where is the displacement, is the initial velocity (before applying the brakes), is the final velocity (which is zero since the car came to a stop), and is time. We can now plug in the known values into the above equation and solve for the initial velocity:

In order to know if the driver was speeding, we must convert this speed and/or the speed limit in the same units. Let's put this speed in kilometers per hour.

As we can see, the driver was exceeding the speed limit and was thus speeding.

A trick that I often use is if something is speeding up, the acceleration and velocity point in the same direction. If something is slowing down. the acceleration and velocity point opposite directions. Since the elevator is speeding up, the acceleration must point in the same direction as it's velocity: downwards.