1.
If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B?
Correct Answer
D. Same magnitude, but must be in opposite directions
Explanation
If two vectors A and B add up to zero (A + B = 0), it means that they cancel each other out. This can only happen if the magnitude (or length) of both vectors is the same. However, since the vectors are in opposite directions, their directions must be opposite as well. Therefore, the vectors A and B must have the same magnitude but must be in opposite directions.
2.
Given that A + B = C, and that lAl ^{2} + lBl^{ 2} = lCl ^{2}, how are vectors A and B oriented with respect to each other?
Correct Answer
A. They are perpendicular to each other
Explanation
Remember the Pythagorean theorem requires that it be a right triangle.
3.
If each component of a vector is doubled, what happens to the angle of that vector?
Correct Answer
C. It does not change
Explanation
The magnitude of the vector clearly doubles if each of its components is doubled. But the angle of the vector is given by tan θ = 2y/2x, which is the same as tan θ = y/x (the original angle).
4.
A certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector, in a standard x-y coordinate system?
Correct Answer
E. 45°
Explanation
The angle of the vector is given by tan θ = y/x. Thus, tan θ = 1 in this case if x and y are equal, which means that the angle must be 45°.
5.
You are adding vectors of length 20 and 40 units. What is the only possible resultant magnitude that you can obtain out of the following choices?
Correct Answer
C. 37
Explanation
The minimum resultant occurs when the vectors are opposite, giving 20 units. The maximum resultant occurs when the vectors are aligned, giving 60 units. Anything in between is also possible, for angles between 0° and 180°.
6.
A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?
Correct Answer
D. It falls right back into the cart
Explanation
In the frame of reference of the cart, the ball only has a vertical component of velocity. So it goes up and comes back down. To a ground observer, both the cart and the ball have the same horizontal velocity, so the ball still returns into the cart.
7.
You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will:
Correct Answer
B. Remain vertically under the plane while falling
Explanation
Both the plane and the package have the same horizontal velocity at the moment of release. They will maintain this velocity in the x-direction, so they stay aligned. There is no acceleration in the x-direction in this case.
8.
From the same height (and at the same time), one ball is dropped and another ball is fired horizontally. Which one will hit the ground first?
Correct Answer
C. They both hit at the same time
Explanation
Both of the balls are falling vertically under the influence of gravity. They both fall from the same height. Therefore, they will hit the ground at the same time. The fact that one is moving horizontally is irrelevant – remember that the x and y motions are completely independent !!
9.
A projectile is launched from the ground at an angle of 30^{o}. At what point in its trajectory does this projectile have the least speed?
Correct Answer
B. At the highest point in its flight
Explanation
The speed is smallest at the highest point of its flight path because the y-component of the velocity is zero.
10.
The spring-loaded gun can launch projectiles at different angles with the same launch speed. At what angle should the projectile be launched in order to travel the greatest distance before landing?
Correct Answer
C. 45°
Explanation
A steeper angle lets the projectile stay in the air longer, but it does not travel so far because it has a small x-component of velocity. On the other hand, a shallow angle gives a large x-velocity, but the projectile is not in the air for very long. The compromise comes at 45°, although this result is best seen in a calculation of the “range formula” as shown in the textbook.
11.
You’re on the street, trying to hit a friend with a water balloon. He sits in his dorm room window above your position. You aim straight at him and shoot. Just when you shoot, he falls out of the window! Does the water balloon hit him??
Correct Answer
A. Yes, it hits
Explanation
Both your friend and the water balloon still fall with the same acceleration -- g = -9.8 !!
12.
It was reported in World War I that a pilot flying at an altitude of 2 km caught in his bare hands a bullet fired at the plane! According to what you know about kinematics and the acceleration of gravity, could this be possible?
Correct Answer
A. Yes! But very unlikely.
Explanation
Remember the bullet's vertical velocity goes to zero before it starts to return to earth. If this is happening around the height of the plane the pilot could grab it. This would of course take lots of luck and extraordinary timing. But still physically possible.