Vector Ventures
Vector v points East. What direction does −v point?
West
A student claims: "Velocity is change in position over change in time, so Δv = Δx/Δt"
Why is this incorrect?
Because Δv and v are not the same thing! The equation should be v = Δv/Δt
A bicyclist starts at point P and travels around a triangular path that takes her through points Q and R before returning to point P. What is the magnitude of her net displacement for the entire trip? [Figure: Motion Diagram #1]
Zero! She starts and ends at the same location!
Two objects move for the same amount of time in the same direction. One travels twice as far as the other. Which object has the greater speed?
The object that traveled twice as far has greater velocity because velocity is displacement divided by time. A larger displacement in the same amount of time means a larger velocity.
The acceleration due to gravity near Earth's surface is approximately _________.
9.81 m/s2 downward
(-9.81 is also an acceptable answer)
A velocity vector has components:
vx = 3 m/s and vy = 4 m/s
What is the magnitude of the velocity?
The magnitude of the velocity is 5 m/s.
You can find this using the Pythagorean theorem because the velocity components form a right triangle with sides of 3 m/s and 4 m/s.
c2 = a2 + b2 = 9 + 16 = 25. If c2 = 25 then c = 5.
A student claims: "Acceleration always means speeding up."
Why is this incorrect?
Because slowing down is also a form of acceleration. Slowing down is negative acceleration. (In physics, we don't use the term deceleration exactly because of this common confusion!)
Describe the motion shown in Motion Diagram #2
The object speeds up at first and then slows down.
If the velocity of a car is non-zero, can the acceleration of the car be zero? Explain.
Yes. If the velocity of a car is non-zero (either positive or negative) but does not change over time, then the acceleration is zero.
Mathematically, acceleration is defined as the change in velocity divided by the change in time: a = Δv/Δt,
acceleration is zero only when Δv = 0, meaning the velocity does not change. The velocity can be any value as long as it stays constant. For example, if the velocity is always 5 m/s, then Δv = v_f - v_i = 5 - 5 = 0.
Instantaneous velocity is always _____ to the path of motion.
Instantaneous velocity is always tangent to the path of motion.
Vector A and and unknown vector B combine to project vector R. Which choice best represents the unknown vector B? [see figure VectorVentures300]
Choice 3 because vectors add tip to tail
A student claims: “If velocity is zero, acceleration must also be zero.”
Why is this incorrect?
v and Δv are not the same thing. This student is confusing these two concepts.
v = Δx/Δt, so if Δx is zero then the velocity must also be zero.
a = Δv/Δt, so Δv is zero then the acceleration must also be zero.
The correct statement would be "If the change in velocity is zero, acceleration must also be zero."
In Motion Diagram #3, which image depicts a car speeding up? The vector arrows represent velocity.
E because the velocity vectors are constantly increasing. This means that velocity is in the same direction as acceleration (e.g., the car is speeding up).
An object moves to the right. Acceleration points to the left. What must be happening to the object’s speed Explain.
The object must be slowing down. If the velocity and acceleration vectors point in opposite directions, the object slows down. If they point in the same direction, the object speeds up.
A ball is thrown upward. At the highest point, what is the acceleration?
9.81 m/s2 pointing downward!
Another acceptable answer would be -9.81 m/s2, because in projectile motion the downward direction is often defined as negative in the coordinate system.
Given the following two vectors A and and B, which of the following pictures correctly represents how you’d combine those vectors to find A - B? (Note: the red vector is the resulting vector in each case.) [Figure VectorVentures400]
(3) because you need to flip B and add that to A using the tip to tail method.
A student claims: "If a ball is thrown vertically upwards, it has a positive acceleration as it goes up and a negative acceleration as it is going down. The acceleration at the very top of the motion is zero."
Why is this incorrect?
Velocity and acceleration are being confused here. What the student has actually described is velocity, not acceleration.
If we choose a coordinate system where up is positive and down is negative, the motion works like this: as the ball moves upward, its velocity is positive. At the highest point, the velocity is zero. As the ball moves downward, the velocity becomes negative.
However, the acceleration is constant the entire time. Because the ball is in free fall, the acceleration is always −g, pointing downward.
In Motion Diagram #4, tennis balls A and B roll down two different ramps. Both ramps have exactly the same slope, but one ramp is twice as long than the other. How do the accelerations of tennis balls A and B compare?
They are the same!
+ is defined as down the ramp. a = Δv/Δt.
Initial point is at t = 0, final point is when the ball reaches the bottom of each ramp
Ball A: a = (16 - 0) / (4 - 0) = 16/4 = 4 m/s2
Ball B: a = (12 - 0) / (3 - 0) = 12/3 = 4 m/s2
Can an object have a negative acceleration and be speeding up? If so, describe a possible physical situation and a corresponding coordinate system. If not, explain why not.
Yes, an object can have negative acceleration and still be speeding up. This happens when both the velocity and the acceleration point in the negative direction.
In that case, the acceleration is acting in the same direction as the velocity, which causes the object’s speed to increase, even though both values are negative.
During projectile motion, which of the following quantities stay constant and which change?
Quantities: displacement, speed, vertical velocity, horizontal velocity, instantaneous velocity, acceleration
Why?
Stay the same: horizontal velocity, acceleration
Change: displacement, speed, vertical velocity, instantaneous velocity
Gravity causes a constant downward acceleration g during projectile motion. Because this acceleration points only downward, it only changes the vertical velocity. The horizontal velocity stays constant.
Displacement changes as the object moves. Instantaneous velocity changes because its direction changes along the curved path. Speed changes because it depends on the magnitude of the velocity.
Vectors A, B, and C are drawn to scale in the figure below. Use the vectors to find R = 3A - 2B + C. (Note: yellow is the resultant vector.) [Figure VectorVentures500]
(2) is the correct vector sum for R = 3A - 2B + C. Note that the order the tip to tail vectors are in is R = C + 3A - 2B, but that is equivalent because addition can be done in any order (e.g., we can move 3A and C around without changing the final answer R).
A student claims: "Acceleration must always point in the same direction as motion."
Why is this incorrect?
Acceleration does not have to point in the same direction as an object’s motion. The direction of motion is the same as the direction of the velocity.
If acceleration points in the same direction as the velocity, the object speeds up (for example, when a car accelerates forward after a stoplight turns green).
If acceleration points in the opposite direction of the velocity, the object slows down (for example, when a moving car brakes as it approaches a red light).
In Motion Diagram #5, carts A and B move along a horizontal track. The motion diagram shows the locations of the carts at instants 1-5 separated by equal time intervals. Is the speed of cart B greater than, less than, or equal to the speed of cart A at instant 2? Explain.
The speed of B is less than the speed of A at instant 2.
Let’s assume, just for simplicity, that each time interval is 1 second. This does not change the answer, but it makes the math easier to think about.
Cart A moves at a constant speed of 4 cm every 1 second, so its speed is always 4 cm/s.
Cart B is accelerating. Between Instants 1 and 2, it travels 1 cm, so its average speed during that interval is 1 cm/s. Between Instants 2 and 3, it travels 2 cm, so its average speed during that interval is 2 cm/s.
That means B’s speed at Instant 2 must be somewhere between 1 cm/s and 2 cm/s. Since that is still less than Cart A’s speed of 4 cm/s, Cart A must be traveling faster than Cart B at Instant 2.
Four objects (A, B, C, and D) travel from left to right. There are equal time intervals between each dot. Rank the objects according to of their accelerations (consider left pointing accelerations as negative).
- <------------> +
A: • • • • • • • • •
B: • • • • •
C: • • • • • • •
D: • • • • • • •
From smallest to largest acceleration: D, B, A, C
Object D has an acceleration in the negative direction, so it has the smallest acceleration.
Object B has no acceleration (its spacing between dots is constant), so a = 0, making it the second smallest.
Objects A and C both have positive accelerations. We can compare their accelerations by looking at how quickly the distances between successive dots increase. In C, the spacing between the dots increases faster than in A, meaning the velocity is increasing more quickly.
Therefore, C has a larger positive acceleration than A.
The figure shows a ball undergoing projectile motion. At each point on the ball's path, three different vectors are drawn that provide information about velocity and acceleration. Given what you know about projectile motion, what do each of these three vector colors represent?
Purple: vertical component of velocity
Red: horizontal component of velocity
Yellow: acceleration due to gravity
The purple vectors point only in the y-direction, and the red vectors point only in the x-direction. Together, they add to form a vector that is tangent to the trajectory at each point.
The yellow vectors also point only in the vertical direction, but they do not change in magnitude throughout the motion. This shows they represent the constant acceleration due to gravity, which is the same everywhere during projectile motion.