Elastic and Inelastic Collisions AP Physics C Mechanics FRQ – Exam Style Questions etc.
Elastic and Inelastic Collisions AP Physics C Mechanics FRQ
Unit 4: Linear Momentum
Weightage : 15-25%
QUESTION
Scientists have created a new type of lightweight foam and are performing experiments to investigate the
properties of the foam. The mass of Cart A is 1000 kg, and the mass of Cart B is 2000 kg. A piece of foam with
negligible mass is attached to the front of Cart A, as shown. Cart A moves with a constant speed toward Cart B,
which is initially at rest. At time t = 0 s, the foam connected to Cart A makes contact with Cart B. The foam
remains in contact with Cart B for 0.5 s, after which the carts separate and both carts move with constant
velocities.
(a) The graph shows the velocity v of Cart A as a function of time t for the time interval when the foam and
Cart B are in contact.
i. What feature(s) of the graph could be used to estimate the displacement of Cart A during the collision?
ii. Using the information shown in the graph, determine the speed of Cart B at t = 0.5 s.
iii. On the following grid, draw a smooth curve of the velocity of Cart B as a function of time.
(b) For 0 ≤ t ≤ 0.50 s, the velocity v of Cart A can be described by the function v(t) = 64t³ − 48t² + 5.
i. Calculate the magnitude of the maximum net force acting on Cart A during this interval.
ii. On the following grid, draw a smooth curve of the magnitude of the force acting on Cart A as a function
of time. Clearly indicate the value of the maximum force on the vertical axis.
The foam is removed from the front of Cart A, and the experiment is repeated. The carts collide, with both Cart A
and Cart B having the same initial and final velocities as in the original collision. The time intervals during which
the carts are in contact are different in the collision with the foam and the collision without the foam. In the
collision without the foam, Cart A is in contact with Cart B for a shorter duration than in the original collision.
when the foam was present.
For the original collision when the foam is present, the magnitude of the average net force exerted on Cart B is $f_{1}$.
For the collision without the foam, the magnitude of the average net force exerted on Cart B is $f_{2}$.
(c) What is the relationship between the magnitude of the average net force $f_{1}$ exerted on Cart B for the collision
with the foam and the magnitude of the average net force $f_{2}$ exerted on Cart B for the collision without the foam?
$ F_{1}> F_{2}$ $F_{1} < F_{2}$ $F_{1} < F_{2}$
Justify your answer.
▶️Answer/Explanation
Ans:-
(a)(i) For stating that the area bounded by the curve is the displacement
(a)(ii) For indicating momentum is conserved
(a)(iii) For a graph that starts at (0,0) and ends at (0.5,2) or value consistent with (a)(ii)
(b)(i) For indicating the derivative of the velocity function is the acceleration of the cart
(b)(ii) For a curve that increases and then decreases in value that is only concave down
(c) For selecting $F_{1}$<$F_{2}$ with an attempt at a relevant justification
Example Solution
Since the initial and final velocities are the same for both collisions, ∆p is the same for both
collisions; as a result, the impulse is the same for both collisions. So, if ∆t is smaller, $F_{avg}$ is larger.
Question
1. Blocks A and B of masses 2m and m, respectively, are arranged in a setup consisting of a ramp that makes an angle q with a smooth horizontal table and an ideal spring of spring constant k fixed to a wall, as shown. Block A is held at rest a distance D up the ramp, and Block B is at rest on the horizontal table. The coefficient of kinetic friction between Block A and the rough ramp is \(\mu \) in the region of length D, and there is negligible friction between the blocks and the smooth table.
At time t = 0, Block A is located at horizontal position x = 0 and is released from rest. After the block is released, the following occurs.
- At time t =\( t_{1}\) , Block A has traveled a distance D down the ramp, has transitioned to the table, and is moving with speed v at x = \(x_{1}\) .
- At time t = \(t_{2} \), Block A is at x = \(x_{2} \) when it collides with and sticks to Block B.
- At time t = \(t_{3} \), the combined blocks A and B are at x = \(x_{3} \) when they collide with and stick to the spring in its equilibrium position.
- At time t = \(t_{4} \), the combined blocks A and B are instantaneously at rest and the spring is compressed a distance \(x_{c} \) from its equilibrium position.
(a) For parts (a)(i) and (a)(ii), express your answer in terms of m, \(\Theta \), D, \(\mu \), \(x_{c}\), and physical constants, as appropriate.
i. Derive an expression for the speed v of Block A at time \(t_{1} \)
ii. Derive an expression for the spring constant k of the spring.
(b)
i. On the following axes, sketch a graph of the magnitude of the momentum \(p_{A} \) of Block A as a function of time t from t = 0 to \(t_{4} \).
ii. Use principles of forces to justify the graph drawn in part (b)(i) for the time interval t t = \(t_{3} \) to t = \(t_{4} \). Explicitly reference features of the shape of the graph you drew in part (b)(i).
For times t > \(t_{4} \), the two-block-spring system oscillates with period \(T_{O} \). The procedure is then repeated using a new ramp, where there is negligible friction between Block A and the ramp.
(c) Indicate how the new period of oscillation \(T_{N} \) in the procedure that uses the new ramp compares with the period of oscillation \(T_{O} \) from the original procedure.
_____ \(T_{N} \) > \(T_{O} \). _____ \(T_{N} \) < \(T_{O} \) _____ \(T_{N} \) = \(T_{O} \)
Briefly justify your answer.
▶️Answer/Explanation
1(a)(i)Example Solution
1(a)(ii)Example Solution
1(b)(i)Example Response
1(b)(ii)Example Response
The momentum of Block A decreases between \(t_{3} \) and \(t_{4} \) because the spring exerts a force on the blocks in the opposite direction of the velocity of the blocks, causing the blocks to slow to a stop. The spring force increases the more the spring compresses, so the momentum of Block A decreases at an increasing rate, which is shown in the slope of the curve becoming steeper with time.
1(c)Example Response
Repeating the experiment on a smooth ramp will only affect the compression distance of the spring. The period of oscillation of a spring-block system depends only on mass and the spring constant, therefore the period of oscillation will not change.
Question
A pendulum of length L consists of block 1 of mass 3M attached to the end of a string. Block 1 is released from rest with the string horizontal, as shown above. At the bottom of its swing, block 1 collides with block 2 of mass M, which is initially at rest at the edge of a table of height 2L. Block 1 never touches the table. As a result of the collision, block 2 is launched horizontally from the table, landing on the floor a distance 4L from the base of the table. After the collision, block 1 continues forward and swings up. At its highest point, the string makes an angle θMAX to the vertical. Air resistance and friction are negligible. Express all algebraic answers in terms of M, L, and physical constants, as appropriate.
(a) Determine the speed of block 1 at the bottom of its swing just before it makes contact with block 2.
(b) On the dot below, which represents block 1, draw and label the forces (not components) that act on block 1 just before it makes contact with block 2. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. Forces with greater magnitude should be represented by longer vectors.
(c) Derive an expression for the tension FT in the string when the string is vertical just before block 1 makes contact with block 2. If you need to draw anything other than what you have shown in part (b) to assist in your solution, use the space below. Do NOT add anything to the figure in part (b). For parts (d)–(g), the value for the length of the pendulum is L = 75 cm.
(d) Calculate the time between the instant block 2 leaves the table and the instant it first contacts the floor.
(e) Calculate the speed of block 2 as it leaves the table.
(f) Calculate the speed of block 1 just after it collides with block 2.
(g) Calculate the angle θMAX that the string makes with the vertical, as shown in the original figure, when block 1 is at its highest point after the collision.
Answer/Explanation
Ans:
(a)
\(mgh = \frac{1}{2}mv^{2}\)
10 (L) = .sv2 v2 = 20L
\(V=\sqrt{20L}=2\sqrt{SL}\)
(b)
(c)
FT = ma
FT – Fg = 3M.
(d)
Xy= Xoy + Voy t+ 1/2 at2
-2L = 0 + 0 t+ 1/2 (-10) t2
-1.5 m = -5t2
3 = t2
\(t = \sqrt{\frac{3}{10}}=0.548 seconds\)
(e)
\(V = \frac{\Delta x}{\Delta t}= \frac{4L}{\sqrt{\frac{3}{10}}}=\frac{3}{\sqrt{\frac{3}{10}}}=5.477 m/s\)
(f)
P1 + P2 = P1f + P2f
m1v1 + m2v2 = m1v1f + m2v2f
\((3m)\left ( \sqrt{20(.70)} \right )+(m)(0)=(3m)(x)+(m)(5.477)\)
11.619 + 0 = 3x + 5.477
6.142 = 3x
x = 2.047 m/s
(g)
\(\frac{1}{2}3MV^{2}=3Mgh\)
\(\frac{1}{2}(2.047)^{2}=10 h\)
H = 0.210 m
\(cos(\theta )=\frac{A}{H}\)
\(\theta = -cos^{-1}\left ( \frac{A}{H} \right )\)
\(\theta = cos^{-1}\left ( \frac{.75-H}{.75} \right )=43.897^{0}\)