Watermark

Vardaan Learning Institute

vardaanlearning.com | 9508841336
Challenger Practice Test: Motion in 1D (Level 3)
Student Name: ____________________________________ Class: 11th (JEE Adv/NEET) Subject: Physics
Section A: Advanced Kinematics & Complex Trajectories
1.
A point moves in a straight line so that its displacement $x$ at time $t$ is given by $x^2 = t^2 + 1$. Show that its acceleration is inversely proportional to $x^3$. Determine the constant of proportionality.
2.
A particle moves along the $x$-axis. Its coordinate is given by $x = 10 + 8t - 3t^2$. Another particle moves along the $y$-axis with coordinate $y = 5 - 8t^3$. Find the magnitude of the relative velocity of the second particle with respect to the first at $t = 1 \text{ s}$.
3.
An ant crawls on a vertical cylinder of radius $R$ and height $H$. It starts from the bottom edge and reaches the diametrically opposite top edge. If it takes exactly two complete helical turns to reach the top, find the magnitude of its net displacement and the total distance covered.
4.
A particle moves such that its position vector is $\vec{r} = (A\sin(\omega t))\hat{i} + (A\cos(\omega t))\hat{j} + (Bt)\hat{k}$. Find the ratio of the magnitude of its velocity to the magnitude of its acceleration.
5.
A person walks on a horizontal plane. First he walks $a$ meters due East, then $2a$ meters due North, then $3a$ meters due West, then $4a$ meters due South, and so on in a spiral pattern. What is the magnitude of his displacement after 10 such distinct segments?
6.
If the displacement of a particle is given by $s = a e^{\alpha t} + b e^{-\beta t}$, where $a, b, \alpha, \beta$ are positive constants, prove that the acceleration is always positive and proportional to the original displacement only if $\alpha = \beta$.
7.
The position of a particle is given by $x(t) = t^3 - 6t^2 + 9t + 5$. Find the total distance travelled by the particle in the interval $t=0$ to $t=4 \text{ s}$. Be careful to locate any turning points.
8.
A point moves such that its distance from the origin at time $t$ is $r = t^2 e^{-t}$. Find the time at which the velocity of the point away from the origin is maximum.
Section B: Variable Rates & Average Velocity Analysis
9.
A particle covers a distance $L$. It travels the first half of the distance with speed $v_1$. The remaining half is covered with speed $v_2$ for half the time and with speed $v_3$ for the other half of the time. Find the average speed of the particle for the entire journey.
10.
A car moves along a straight line whose speed varies with distance $s$ as $v = \alpha \sqrt{s}$, where $\alpha$ is a constant. Find the average velocity of the car over a total distance $S$.
11.
A point traversing a straight line moves with a constant velocity $v_1$ for $1/n^{\text{th}}$ of the total distance, then with velocity $v_2$ for the next $1/n^{\text{th}}$ of the distance, and so on up to $v_n$. Prove that the average velocity is the harmonic mean of the individual velocities.
12.
A particle's velocity is given by $v(t) = v_0 e^{-bt}$ where $v_0$ and $b$ are positive constants. Find the average velocity of the particle between $t=0$ and the time when its velocity reduces to $v_0/2$.
13.
An object travels continuously in one direction. During the first $N$ seconds, its velocity at time $t$ is given by $v = k t^2$. Calculate the ratio of its average velocity during the first $N$ seconds to its instantaneous velocity at $t = N$.
14.
A particle starts from rest and moves with acceleration $a$ which varies with time $t$ as $a = kt$, where $k$ is a constant. Find the average velocity of the particle during the time interval it takes to cover a distance $S$.
15.
The speed of a train varies as $v = v_0 \sin(\pi t / T)$ for $0 \le t \le T$. Find the average speed of the train for this duration and compare it to the maximum speed $v_0$.
16.
A body moves in a straight line with a velocity varying as the square of the time. If it covers a distance $x$ in time $t$, and $y$ in the next time $t$, prove that $y = 7x$.
Section C: Differential Equations in Kinematics
17.
The deceleration experienced by a moving motor boat after its engine is cut off is given by $dv/dt = -kv^3$, where $k$ is a constant. If $v_0$ is the magnitude of the velocity at cut-off, find the velocity as a function of time $t$ and distance $x$.
18.
A particle of mass $m$ moves in a straight line under a retarding force which gives it an acceleration $a = -k \sqrt{v}$, where $v$ is velocity. If the initial velocity is $u$, find the total distance covered by the particle before coming to rest.
19.
A particle moves along a straight line such that its velocity $v$ depends on displacement $x$ as $v = \beta x^{-2n}$. Find the acceleration of the particle as a function of time, given that $x(0) = 0$.
20.
The acceleration of a particle is defined by the relation $a = -k/x^2$. The particle starts with no initial velocity at $x = x_0$. Determine the velocity of the particle when $x = x_0/2$.
21.
A point moves rectilinearly with deceleration whose modulus depends on the velocity $v$ of the particle as $w = a\sqrt{v}$, where $a$ is a positive constant. At the initial moment, the velocity of the point is equal to $v_0$. What distance will it traverse before it stops? What time will it take to cover that distance?
22.
A point mass moves along the $x$-axis. At time $t=0$, $x=0$ and $v=v_0$. The acceleration is given by $a = -cx^3$ where $c$ is a positive constant. Find the maximum distance the point can move along the positive $x$-axis.
23.
For a particle moving in a straight line, the relation between time $t$ and position $x$ is $t = \alpha x^2 + \beta x$. Show that the retardation of the particle is proportional to the cube of its instantaneous velocity. Evaluate the constant of proportionality.
24.
An object moving in a straight line experiences an acceleration $a = A \cos(\omega t)$ where $A$ and $\omega$ are constants. At $t=0$, $x=0$ and $v=v_0$. Find the position $x(t)$. Under what condition will the particle never reverse its direction?
25.
Velocity $v$ of a particle moving along a straight line varies with distance $x$ as $v = a - bx$, where $a$ and $b$ are positive constants. Find the time at which the velocity becomes zero.
26.
A particle is moving such that its velocity is $v = c \ln(x)$. Find the acceleration as a function of $x$. Also find the velocity as a function of time if $x(0) = x_0$.
Section D: Complex Systems under Uniform Acceleration
27.
Two cars A and B are simultaneously travelling on a straight road in the same direction. Car A starts from rest with acceleration $a_1$. Car B is travelling with constant velocity $u$. Find the maximum distance by which Car A lags behind Car B, and the time when A overtakes B.
28.
A train of length $l$ moves with a constant acceleration $a$. A person standing on a platform notes that the front of the train passes him with velocity $u$ and the rear with velocity $v$. What was the velocity of the train when exactly half of its length had passed the person?
29.
A particle starts from rest and moves with uniform acceleration. It covers $x$ meters in the $p^{\text{th}}$ second and $y$ meters in the $q^{\text{th}}$ second. Prove that its acceleration is $\frac{2(x-y)}{p-q}$.
30.
A car starting from rest accelerates at the rate $f$ through a distance $S$, then continues at constant speed for time $t$, and then decelerates at the rate $f/2$ to come to rest. If the total distance traversed is $15S$, then determine $S$ in terms of $f$ and $t$.
31.
Two particles start moving simultaneously from the same point in the same direction. The first particle has initial velocity $u$ and uniform acceleration $a_1$. The second particle has initial velocity $0$ and uniform acceleration $a_2 > a_1$. Find the maximum separation between them before the second overtakes the first.
32.
A driver travelling at speed $V$ suddenly sees a wall at a distance $d$ directly in front of him. His reaction time is $t_r$. Find the minimum retardation required to avoid a collision.
33.
If a body loses half of its velocity on penetrating $3 \text{ cm}$ in a wooden block, how much will it penetrate more before coming to rest, assuming uniform resistance?
34.
Two trains are moving on the same track in opposite directions at speeds $u_1$ and $u_2$. They are separated by distance $D$ when the drivers apply their brakes, producing retardations $a_1$ and $a_2$. Prove that they will avert a collision if $D > \frac{u_1^2}{2a_1} + \frac{u_2^2}{2a_2}$.
35.
A point moves with uniform acceleration. In the $11^{\text{th}}$ and $15^{\text{th}}$ seconds it covers distances of $24 \text{ m}$ and $32 \text{ m}$ respectively. Find its initial velocity, acceleration, and the distance covered in the first $10$ seconds.
36.
An anti-aircraft gun can fire bullets at $500 \text{ m/s}$. If a fighter plane is flying horizontally at an altitude of $2 \text{ km}$ with a speed of $1000 \text{ km/h}$, what should be the angle of elevation of the gun to hit the plane if the gun is fired exactly when the plane is directly overhead? (Assume $g$ is constant, ignore air friction).
Section E: Complex Motion Under Gravity
37.
A parachutist bails out from an aeroplane and after dropping through a distance of $40 \text{ m}$, he opens the parachute and decelerates at $2 \text{ m/s}^2$. If he reaches the ground with a speed of $2 \text{ m/s}$, how long was he in the air? At what height did he bail out? ($g=9.8 \text{ m/s}^2$)
38.
A ball is dropped from the roof of a tower of height $h$. The total distance covered by it in the last second of its motion is equal to the distance covered by it in the first three seconds. What is the height $h$ of the tower? ($g=10 \text{ m/s}^2$)
39.
Two bodies begin to fall from the same height, but the second falls $T$ seconds after the first. Find the time (after the first body starts falling) at which the distance between the bodies equals $L$.
40.
A stone is dropped from a balloon ascending with a constant velocity of $12 \text{ m/s}$. The stone reaches the ground in $4 \text{ s}$. Find the height of the balloon at the instant the stone was dropped and the velocity of the stone right before it strikes the ground. ($g=9.8 \text{ m/s}^2$)
41.
A ball is thrown vertically upward with speed $u$. If it is at a certain height $h$ at two different times $t_1$ and $t_2$, prove that $t_1 + t_2 = \frac{2u}{g}$ and $t_1 t_2 = \frac{2h}{g}$.
42.
An elevator car is moving upward with uniform acceleration $a$. A passenger drops a coin from a height $h$ above the floor of the elevator. How long will the coin take to strike the floor?
43.
A particle is thrown vertically upwards from the surface of the earth. Let $T_P$ be the time taken by the particle to travel from a point $P$ above the earth to its highest point and back to the point $P$. Similarly, let $T_Q$ be the time taken by the particle to travel from another point $Q$ above the earth to its highest point and back to $Q$. Prove that the distance $PQ = \frac{g(T_P^2 - T_Q^2)}{8}$.
44.
A stone is dropped from the top of a cliff. It is seen to pass a window $1.5 \text{ m}$ high in $0.1 \text{ s}$. How far above the top of the window was the stone dropped? ($g=10 \text{ m/s}^2$)
45.
Water drops are falling from a nozzle of a shower onto the floor, from a height of $9.8 \text{ m}$. The drops fall at a regular interval of time. When the first drop strikes the floor, at that instant, the third drop begins to fall. Locate the position of the second drop from the floor when the first drop strikes the floor.
46.
A body falling freely from a given height $H$ hits an inclined plane in its path at a height $h$. As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. For what value of $h / H$ will the body take the maximum time to reach the ground?
Section F: Non-Inertial Frames & Relative Velocity
47.
Two straight railway tracks are at right angles to each other. At time $t=0$, a train A on one track is at distance $a$ from the intersection and moving towards it with speed $u$. Another train B on the other track is at distance $b$ from the intersection and moving away from it with speed $v$. Find the shortest distance between them and the time when they are closest.
48.
A passenger is running at maximum speed of $8 \text{ m/s}$ to catch a train. When he is distance $d$ from the door, the train starts from rest with a constant acceleration of $1 \text{ m/s}^2$. What is the maximum value of $d$ so that he can just board the train?
49.
Ship A is sailing towards the north-east with velocity $\vec{v} = 30\hat{i} + 50\hat{j} \text{ km/hr}$. Ship B is at a distance of $80 \text{ km}$ east and $150 \text{ km}$ north of Ship A and is sailing towards the west at $10 \text{ km/hr}$. A will be at minimum distance from B in what time?
50.
On a two-lane road, car A is travelling with a speed of $36 \text{ km/h}$. Two cars B and C approach car A in opposite directions with a speed of $54 \text{ km/h}$ each. At a certain instant, when the distance AB is equal to AC, both being $1 \text{ km}$, B decides to overtake A before C does. What minimum acceleration of car B is required to avoid an accident?
51.
A lift starts ascending with constant acceleration $a$. A passenger throws a ball vertically up with a velocity $v_0$ relative to the lift. Find the time of flight of the ball and its maximum height relative to the lift floor.
52.
A conveyer belt moves with a constant velocity $v_0$. A block is gently placed on it. The coefficient of kinetic friction between the block and belt is $\mu_k$. Find the distance the block slides on the belt before coming to rest relative to the belt, and the total distance moved by the block relative to the ground.
53.
Two particles $P$ and $Q$ move in a straight line $AB$. $P$ starts from $A$ with velocity $u_1$ and acceleration $f_1$. Simultaneously $Q$ starts from $B$ with velocity $u_2$ and acceleration $f_2$ towards $A$. If they pass one another at the middle point of $AB$ and arrive at the other ends of $AB$ with equal velocities, prove that $(u_1 + u_2)(f_1 - f_2) = 8(f_1 u_2 - f_2 u_1)$.
54.
An open elevator is ascending with constant acceleration $a=1.2 \text{ m/s}^2$. A boy in the elevator throws a ball vertically upwards with a speed $v_0=5 \text{ m/s}$ relative to the elevator. If the initial height of the ball from the elevator floor is $2 \text{ m}$, find how long the ball takes to hit the floor. ($g=9.8 \text{ m/s}^2$)
Section G: Advanced Graphical Interpretations
55.
The graph of $1/v$ versus displacement $x$ is a straight line passing through the origin with a slope $k$. Determine the relation between acceleration $a$ and velocity $v$.
56.
An acceleration-displacement ($a-x$) graph for a particle moving in a straight line is given as a straight line passing through $(0, a_0)$ and $(x_0, 0)$. If the particle starts from rest at $x=0$, find its velocity as a function of $x$.
57.
The velocity-displacement ($v-x$) graph of a particle moving in a straight line is a straight line intercepting the $v$-axis at $v_0$ and the $x$-axis at $x_0$. Prove that the acceleration of the particle varies linearly with displacement $x$, and find its maximum magnitude.
58.
A particle's velocity $v$ squared ($v^2$) is plotted against displacement $x$ resulting in a straight line with a positive slope $m$ and y-intercept $C$. What are the physical significances of $m$ and $C$? What is the acceleration of the particle?
59.
A particle starts from rest and undergoes an acceleration $a$ that varies with time $t$ as shown in an $a-t$ graph which is an isosceles triangle with base on the time axis from $t=0$ to $t=2t_0$ and peak $a_0$ at $t=t_0$. Calculate the maximum velocity and total distance covered by the particle in time $2t_0$.
60.
If the position of a particle varies with time as $x = A \sin(\omega t)$, sketch the graphs of $x-t$, $v-t$, $a-t$ and $a-x$. State the nature of the $v-x$ graph.
Answer Key (Level 3 - Short Form)
Q1: $a = 1/x^3$, $k=1$
Q2: $\sqrt{580} \text{ m/s}$ (approx $24.08$)
Q3: Disp: $\sqrt{H^2+4R^2}$, Dist: $\sqrt{H^2+16\pi^2 R^2}$
Q4: $\sqrt{A^2\omega^2 + B^2}/(A\omega^2)$
Q5: $5\sqrt{2}a$
Q6: Proof ($a = \alpha^2 s$)
Q7: $24 \text{ m}$ (Turns at $t=1, 3$)
Q8: $t = 2 - \sqrt{2} \text{ s}$
Q9: $\frac{2v_1(v_2+v_3)}{2v_1+v_2+v_3}$
Q10: $\frac{2}{3}\alpha\sqrt{S}$
Q11: Proof ($\frac{n}{\sum 1/v_i}$)
Q12: $\frac{v_0}{2\ln 2}$
Q13: $1 : 3$
Q14: $(kS^2/6)^{1/3} \times (3/2)$
Q15: $v_{avg} = \frac{2}{\pi}v_0$
Q16: Proof
Q17: $v = \frac{v_0}{\sqrt{1+2kv_0^2 t}}$, $v = \frac{v_0}{1+kv_0 x}$
Q18: $S = \frac{2u^{3/2}}{3k}$
Q19: $a = -2n\beta^2 x^{-4n-1}$
Q20: $v = \sqrt{2k/x_0}$
Q21: $S = \frac{2v_0^{3/2}}{3a}$, $T = \frac{2\sqrt{v_0}}{a}$
Q22: $x_{max} = (\frac{2v_0^2}{c})^{1/4}$
Q23: $a = -2\alpha v^3$
Q24: $v_0 \ge \frac{A}{\omega}$
Q25: $t \to \infty$
Q26: $a = \frac{c^2 \ln x}{x}$
Q27: $D_{max}=\frac{u^2}{2a_1}$, $t=\frac{2u}{a_1}$
Q28: $\sqrt{\frac{u^2+v^2}{2}}$
Q29: Proof
Q30: $S = \frac{1}{72} f t^2$
Q31: $S_{max} = \frac{u^2}{2(a_2-a_1)}$
Q32: $a_{min} = \frac{V^2}{2(d - V t_r)}$
Q33: $1 \text{ cm}$
Q34: Proof
Q35: $u=3\text{m/s}, a=2\text{m/s}^2, S_{10}=130\text{m}$
Q36: $\theta = \sin^{-1}(\frac{1000 \times 5/18}{500})$ No, wait, horizontal aim ahead.
Q37: $t = 16.93\text{s}$, $H = 234\text{m}$
Q38: $125 \text{ m}$ ($t=5\text{s}$)
Q39: $t = \frac{L}{gT} + \frac{T}{2}$
Q40: $H = 30.4\text{m}, v = 27.2\text{m/s}$
Q41: Proof (Quadratic in t)
Q42: $t = \sqrt{\frac{2h}{g+a}}$
Q43: Proof
Q44: $10.125 \text{ m}$
Q45: $7.35 \text{ m}$ from floor
Q46: $h/H = 1/2$
Q47: $S_{min} = \frac{|au - bv|}{\sqrt{u^2+v^2}}$
Q48: $d \le 32 \text{ m}$
Q49: $t = 2.6 \text{ hours}$
Q50: $a_{min} = 1 \text{ m/s}^2$
Q51: $T = \frac{2v_0}{g+a}, H = \frac{v_0^2}{2(g+a)}$
Q52: $S_{rel}=\frac{v_0^2}{2\mu_k g}$, $S_{grnd}=\frac{v_0^2}{2\mu_k g}$
Q53: Proof
Q54: $t = 0.81 \text{ s}$
Q55: $a = -k v^3$
Q56: $v = \sqrt{a_0 x(2 - x/x_0)}$
Q57: $a_{max} = v_0^2/x_0$ at origin
Q58: $m=2a$, $C=u^2$, $a=m/2$
Q59: $V_{max}=a_0 t_0$, $S=a_0 t_0^2$
Q60: Ellipse ($v^2/\omega^2 A^2 + x^2/A^2 = 1$)
Created by Team Vardaan | Powered by Vardaan Comet
Vardaan Learning Institute © 2026 - All Rights Reserved