Speed is defined as the magnitude of velocity. In other words, speed visually represents the size of a velocity vector, but is indepedent of its direction.

Velocity, acceleration, and force are all vector quantities. Each is defined by the magnitude of the measurement as well as its direction of action.

This question tests your understanding of displacement as a vector quantity. The best way to solve it is to create a number line to track the motion of the dog.

Let the initial position of the dog be 0m on our number line:

Then the dog moves three meters to the right, which we can represent as follows:

Then the dog moves seven meters to the left. Note that it is moving seven meters to the left from the position it is at now (3m):

We find that at the end of its motion, the dog's displacement is four meters to the left of its original position, which we write as .

Remember that displacement is the change in position, and not the total distance traveled. We only care about the initial position and the final position, regardless of the path traveled. Positive and negative signs indicate the direction of motion.

This question tests your conceptual understanding of distance as a scalar quantity vs your understanding of displacement as a vector quantity. 

Distance measures the total length that was traveled in a given motion, and does not care about the direction since it is a scalar value. In your day, you traveled 50km on your way to school and 50km on your way back home. In total you traveled 100km, so that is your distance.

Displacement is a vector quantity that measures the change in position. It cares about your final and initial positions, taking into account the direction of the change in position. In this scenario you started and ended your motion exactly at the same position, so overall at the end of the day your car did not change position at all. Therefore your displacement was 0m.

This is a simple question that tests your conceptual understanding of speed as a scalar quantity and velocity as a vector quantity. The motion of the object is quite simple, so you need only to be mindful of the fact that velocity, as a vector, must tell you both the magnitude and direction (how fast it is going and where), while speed only tells you the magnitude (how fast it is going).

Therefore, your speed is  and your velocity is .

Since the acceleration is constant in this problem, we can apply the kinematics given equation to calculate the distance:

First, we need to calculate the acceleration.

Plug in our velocity and time values to find the acceleration.

Now we can return to the kinematics equation and solve for the distance traveled:

First, find the displacement equations for both cars. Car 1 will be the car that is initially stationary; car 2 will be the car traveling with constant velocity.

Car 1:

Car 2:

Now, set those displacement equations equal to each other and solve for .

The accelerating car will catch and pass the car traveling at constant velocity after 4 seconds.

Use the fact that to find the velocity equation, and then solve for when .

Take the derivative of the displacement equation.

Set the velocity equal to zero.

Solve for the time.

Take the second derivative of , to find the acceleration function . Then find the value of where . The fact that it is simply  and not or inside that quantity means that using the Chain Rule becomes decidedly easier.

Set the acceleration fuction equal to zero and solve for the time.

The projectile travels the farthest when the vertical component of its velocity matches the sum of its horizontal component and whatever the wind/friction adds or subtracts. If the wind has no effect, then a 45-degree angle will be the best because the horizontal and vertical components of the velocity will create a right isosceles triangle (remember special triangles).

To solve this problem you need to obtain a function of position with respect to time. For that, you need to understand velocity as the rate of change of displacement with respect to time. In other words, velocity is "how fast" (i.e. how much time it takes) an object changes position (remember displacement is the change in position). This means that velocity is the derivative of displacement with respect to time.

Therefore, to obtain a function of position with respect to time you need to take the antiderivative of the velocity function, so we integrate:

Here,  is a constant that represents the initial position of the particle. We know that the initial position of the particle is 0m, so our function is:

Therefore, after two seconds have passed, we have t = 2s and

Note: we know that position is given in meters since the question specified that velocity is measured in meters per second.