Unit: 1. Kinematics
Weightage : 10-15%
Figure 1 shows a person dropping a coffee filter to the floor. As the filter drops, air resistance causes the acceleration to decrease with time. Figure 2 shows the velocity \(v\) and acceleration \(a\) vectors a short time after the filter is dropped.
Which of the following could correctly show the velocity and acceleration vectors a short time after Figure 2?
Correct Answer: \( \boxed{\mathrm{D}} \)
As the coffee filter falls, two forces act on it:
• Weight \(mg\) downward.
• Air resistance upward.
The net force is
\(F_{\mathrm{net}} = mg – F_{\mathrm{drag}}\),
so the acceleration is
\(a=\dfrac{mg-F_{\mathrm{drag}}}{m}\).
As the filter speeds up, the drag force increases. Therefore, the net downward force decreases, causing the magnitude of the downward acceleration to become smaller.
However, because the acceleration is still downward, the filter continues to gain downward speed. Thus:
• The velocity vector should be longer than before (greater downward speed).
• The acceleration vector should still point downward but be shorter (smaller magnitude).
Only Option (D) satisfies both conditions. Eventually, when the drag force equals the weight, the filter reaches terminal velocity and the acceleration becomes zero.
Vectors \(V_1\) and \(V_2\) shown above have equal magnitudes. The vectors represent the velocities of an object at times \(t_1\) and \(t_2\), respectively. The average acceleration of the object between time \(t_1\) and \(t_2\) is
Correct Answer: \( \boxed{\mathrm{E}} \)
The average acceleration is determined from the change in velocity:
\( \vec{a}_{\mathrm{avg}}=\dfrac{\Delta\vec{v}}{\Delta t} =\dfrac{\vec{V}_2-\vec{V}_1}{\Delta t}. \)
The initial velocity \(\vec{V}_1\) points north of east, while the final velocity \(\vec{V}_2\) points due north. Since the two vectors have equal magnitudes, subtracting \(\vec{V}_1\) from \(\vec{V}_2\) is equivalent to adding the opposite of \(\vec{V}_1\), which points south of west.
The resulting change in velocity has:
• A westward component because the eastward component of \(\vec{V}_1\) is removed.
• A northward component because \(\vec{V}_2\) has a larger northward component than \(\vec{V}_1\).
Therefore, the average acceleration points north of west.
Hence, the correct answer is Option (E).
Starting from rest, a cart moves to the right in a straight line along a track. The motion of the cart is represented by the dots shown, recorded at equal time intervals.
Which of the following best represents the acceleration vectors at the beginning, middle, and end of the motion?
Correct Answer: \( \boxed{\mathrm{B}} \)
In a motion diagram, the spacing between successive dots represents the distance traveled during equal time intervals. Therefore, larger spacing corresponds to greater speed.
At the beginning, the spacing between the dots increases, indicating that the cart is speeding up while moving to the right. Thus, the acceleration points to the right.
In the middle, the dots are approximately equally spaced, indicating nearly constant velocity. Therefore,
\(a = 0\).
At the end, the spacing between the dots decreases while the cart is still moving to the right. This means the cart is slowing down, so the acceleration points to the left, opposite the direction of motion.
Since acceleration is the rate of change of velocity,
\( \vec{a}=\dfrac{\Delta \vec{v}}{\Delta t} \),
the correct sequence is: rightward acceleration, zero acceleration, and leftward acceleration, which corresponds to Option (B).
