The force-momentum relationship is:

Since both force and change in time are located on the left side of the equation, they are inversely proportional. Therefore, if the force doubles, the inverse must be true about the time. This would halve the time necessary for the cars to bump. 

This is a conservation of momentum problem:

 where the subscripts refer to  , and 

Now, let's plug in the relevant information. Remember, the hoverboard is initially stationary, so its velocity is zero. Also, for the final mass, we need to add the two masses, since now Sally is on top of the hoverboard.

Now, all we need to do is solve for the final velocity:

For this question, we're told that a runner begins at rest and travels a certain distance with a constant acceleration. We're asked to determine how much time it takes for this to happen.

Right off the bat, it's important to realize that we can use the kinematics equations because we are told that the acceleration is constant. Now, we need to use an equation that relates distance, time, and acceleration.

Since we know that the person is starting from rest, their initial velocity must be zero. Thus, we can simplify the above expression.

Now, we can go ahead and rearrange this expression to isolate the variable for time.

Lastly, we just need to plug in the values given to us in the question stem in order to solve for our answer.

In this question, we're told that two objects of equal mass eventually collide with each other and then travel together. We're asked to find an expression that tells us the final momentum of the system.

First, we can recognize this as an inelastic collision, which means that momentum is conserved in our system of the two objects but kinetic energy is not. Also, because both objects travel together after the collision, there will be a single final velocity.

Thus, the final momentum of our system of two objects can be expressed as .

At a projectiles maximum height its y-direction velocity is always . It is moving neither up nor down. Because the ball was thrown straight in the air, there is no x-direction component of its velocity to account for.

Gravity is the only force acting on the ball at its maximum height. Though the ball has zero velocity at this point, gravity is still accelerating the ball at . Gravity almost always pulls down, which is a negative direction in the traditional coordinate system so our answer must be negative.

We can calculate the distance traveled by multiplying the velocity of the ball by the reaction time of the batter. However, we will need to first need to get the appropriate units:

Now we can simply multiply this by the reaction time of the batter to get the distance that the ball has traveled by the time her reacts:

Fun fact: the distance from the pitching mound to home plate is about , so the ball is about halfway to the plate by the time the batter begins to react.

There air multiple ways to solve this problem, so don't worry if you attacked this problem from a different angle. This following method avoids using the quadratic formula:

First, we will calculate how long it takes for the ball to travel upward, and then come back down to its original height. 

Since we are neglecting air resistance, we know that our final velocity is equal but in the opposite direction as the initial velocity. Therefore, we get:

At this point, the ball is at it's original height of , but traveling downward at a rate of .

We can then use the following expression to calculate the velocity of the ball as it hits the ground:

Plugging in our values:

Then using our original expression again:

Adding our times together:

For this problem, the following kinematic equation is necessary:

Since the plane starts at rest, we can simplify the equation to

The problem defines the acceleration to be  and the time to be  seconds so