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Level 2 Practice Test: Motion in One Dimension
Student Name: ____________________________________ Class: 11th (JEE/NEET) Subject: Physics
Section A: Position, Path Length & Displacement
1.
A particle moves along a circular path of radius $R$. What is the ratio of the distance to the displacement of the particle when it completes half a revolution?
2.
An athlete completes one round of a circular track of radius $R$ in $40 \text{ s}$. What will be his displacement at the end of $2 \text{ minutes } 20 \text{ s}$?
3.
A boy walks $4 \text{ m}$ east and then $3 \text{ m}$ south. Find the total distance travelled and the magnitude of his net displacement.
4.
A drunkard taking a step of $1 \text{ m}$ in $1 \text{ s}$ takes $5$ steps forward and $3$ steps backward. How long will he take to fall into a pit $13 \text{ m}$ away from his starting point?
5.
Can the displacement of a moving particle be zero? If yes, give an example. Can the distance travelled be zero for a moving particle?
6.
The numerical ratio of displacement to distance for a moving object is always: (a) always less than $1$ (b) always equal to $1$ (c) always more than $1$ (d) equal to or less than $1$.
7.
A wheel of radius $1 \text{ m}$ rolls forward half a revolution on a horizontal ground. The magnitude of the displacement of the point of the wheel initially in contact with the ground is?
Section B: Average Speed & Average Velocity
8.
A car travels the first half of a distance between two places at a speed of $30 \text{ km/hr}$ and the second half of the distance at $50 \text{ km/hr}$. Find the average speed of the car.
9.
A train travels from city A to city B with a constant speed of $10 \text{ m/s}$ and returns back to city A with a constant speed of $20 \text{ m/s}$. Find its average velocity and average speed for the entire journey.
10.
A body covers one-third of its journey with speed $u$, next one-third with speed $v$ and the last one-third with speed $w$. Calculate the average speed of the body during the entire journey.
11.
A person moves with a velocity $v_1$ for time $t_1$ and with a velocity $v_2$ for time $t_2$. Derive the expression for the average velocity if both motions are in the same direction.
12.
A boy runs on a straight road from his home to a market $2.5 \text{ km}$ away with a speed of $5 \text{ km/h}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 \text{ km/h}$. What is the magnitude of average velocity over the interval $0$ to $40 \text{ min}$?
13.
If a particle moves with speed $v_1$ for half the time of travel and $v_2$ for the rest half of the time, what is its average speed?
14.
An object travels $20 \text{ m}$ in $2 \text{ s}$ and then another $16 \text{ m}$ in $2 \text{ s}$. What is the average speed of the object?
15.
Can an object have a constant speed but a variable velocity? Explain with an example from 2D motion, and state if it is possible in strictly 1D motion.
Section C: Calculus in Kinematics (Instantaneous Values)
16.
The position of an object moving along x-axis is given by $x = a + bt^2$ where $a = 8.5 \text{ m}$, $b = 2.5 \text{ m/s}^2$ and $t$ is measured in seconds. What is its velocity at $t = 0 \text{ s}$ and $t = 2.0 \text{ s}$?
17.
The displacement $x$ of a particle varies with time $t$ as $x = 4t^2 - 15t + 25$. Find the position, velocity, and acceleration of the particle at $t = 0$.
18.
The acceleration of a particle is given by $a = 3t^2 + 2t + 2$. If the particle starts with a velocity $v = 2 \text{ m/s}$ at $t = 0$, find the velocity at the end of $2 \text{ s}$.
19.
A particle moves along a straight line such that its displacement at any time $t$ is given by $s = (t^3 - 6t^2 + 3t + 4) \text{ m}$. Find the velocity when the acceleration is zero.
20.
The velocity of a particle depends upon time as $v = A + Bt + Ct^2$. If velocity is in $\text{m/s}$, find the dimensions of $A, B$, and $C$.
21.
Velocity of a particle is given by $v = \alpha \sqrt{x}$, where $x$ is displacement and $\alpha$ is a constant. Find the acceleration of the particle.
22.
The relation between time $t$ and distance $x$ is $t = ax^2 + bx$, where $a$ and $b$ are constants. Express the instantaneous acceleration in terms of velocity $v$.
23.
A particle's velocity is given by $v = t^2 - 4t + 3$. Find the total distance travelled by the particle in the first $4$ seconds. (Hint: Watch for turning points).
24.
The acceleration of a particle varies with position $x$ as $a = -kx$. If the initial velocity at $x=0$ is $v_0$, find the velocity as a function of $x$.
25.
Displacement is given by $x = 2t^3 - 3t^2 + 4t$. Calculate the average acceleration between $t=1 \text{ s}$ and $t=3 \text{ s}$.
Section D: Equations of Kinematics (Uniform Acceleration)
26.
A jet plane starts from rest with an acceleration of $3 \text{ m/s}^2$ and makes a run for $35 \text{ s}$ before taking off. What is the minimum length of the runway and what is the velocity of the jet at take off?
27.
A driver takes $0.20 \text{ s}$ to apply the brakes after he sees a need for it. This is called the reaction time of the driver. If he is driving a car at a speed of $54 \text{ km/h}$ and the brakes cause a deceleration of $6.0 \text{ m/s}^2$, find the distance travelled by the car after he sees the need to put the brakes on.
28.
A bullet moving with a velocity of $200 \text{ m/s}$ penetrates a wooden block and comes to rest after traversing $4 \text{ cm}$ inside it. What acceleration (assumed uniform) is offered by the block?
29.
A particle starting from rest experiences a constant acceleration. It travels a distance $x$ in the first $10 \text{ s}$ and a distance $y$ in the next $10 \text{ s}$. Find the relation between $x$ and $y$.
30.
A car accelerates from rest at a constant rate $\alpha$ for some time, after which it decelerates at a constant rate $\beta$ and comes to rest. If the total time elapsed is $t$, evaluate the maximum velocity acquired.
31.
Derive the equation $S_n = u + \frac{a}{2}(2n - 1)$ for the distance covered by a uniformly accelerating body in the $n^{\text{th}}$ second of its motion.
32.
A body covers $12 \text{ m}$ in the $2^{\text{nd}}$ second and $20 \text{ m}$ in the $4^{\text{th}}$ second. Find its initial velocity and acceleration.
33.
Stopping distance of a car is $d$ when it is moving with speed $v$. If the speed becomes $n$ times, what will be the new stopping distance assuming the same retarding force?
34.
A train of length $L$ crosses a pole with constant acceleration. The front of the train passes the pole with velocity $u$ and the rear of the train passes with velocity $v$. Find the velocity with which the middle point of the train passes the pole.
35.
A car moving with a speed of $40 \text{ km/h}$ can be stopped by applying brakes after at least $2 \text{ m}$. If the same car is moving with a speed of $80 \text{ km/h}$, what is the minimum stopping distance?
Section E: Motion Under Gravity (Free Fall)
36.
A ball is thrown vertically upwards with a velocity of $20 \text{ m/s}$ from the top of a multistorey building. The height of the point from where the ball is thrown is $25.0 \text{ m}$ from the ground. How high will the ball rise? ($g = 10 \text{ m/s}^2$)
37.
Referring to the previous question, how long will it be before the ball hits the ground?
38.
A stone is dropped from a height $h$. It hits the ground with a certain momentum $P$. If the same stone is dropped from a height $100\%$ more than the previous height, what is the percentage change in momentum?
39.
A body falls freely from rest. Show that the distances fallen in successive equal time intervals are in the ratio $1 : 3 : 5 : 7 \dots$ (Galileo's law of odd numbers).
40.
Two balls are dropped from different heights $h_1$ and $h_2$. Find the ratio of the times taken by them to reach the ground.
41.
Water drops fall at regular intervals from a tap $5 \text{ m}$ above the ground. The third drop is leaving the tap at the instant the first drop touches the ground. How high above the ground is the second drop at that instant? ($g = 10 \text{ m/s}^2$)
42.
A balloon is ascending at the rate of $9.8 \text{ m/s}$ at a height of $39.2 \text{ m}$ above the ground when a food packet is dropped from it. After how much time and with what velocity does it reach the ground?
43.
A juggler maintains four balls in motion, making each in turn rise to a height of $20 \text{ m}$ from his hand. With what velocity does he project them and where will the other three balls be at the instant when the fourth one is just leaving his hand? ($g = 10 \text{ m/s}^2$)
44.
A stone falls from a tower and travels $45 \text{ m}$ in the last second of its journey. Find the height of the tower. ($g = 10 \text{ m/s}^2$)
45.
Is the acceleration of a vertically thrown object zero at its highest point? Explain.
46.
A rocket is fired upward from the earth's surface such that it creates an acceleration of $19.6 \text{ m/s}^2$. If after $5 \text{ s}$ its engine is switched off, the maximum height of the rocket from earth's surface would be?
Section F: Relative Velocity in 1D
47.
Two parallel rail tracks run north-south. Train A moves north with a speed of $54 \text{ km/h}$, and train B moves south with a speed of $90 \text{ km/h}$. What is the velocity of B with respect to A?
48.
From the previous question, find the velocity of a monkey running on the roof of train A against its motion (with a velocity of $18 \text{ km/h}$ with respect to the train A) as observed by a man standing on the ground.
49.
A police van moving on a highway with a speed of $30 \text{ km/h}$ fires a bullet at a thief's car speeding away in the same direction with a speed of $192 \text{ km/h}$. If the muzzle speed of the bullet is $150 \text{ m/s}$, with what speed does the bullet hit the thief's car?
50.
Two trains, each $50 \text{ m}$ long, are travelling in opposite directions with velocities $10 \text{ m/s}$ and $15 \text{ m/s}$. The time of their crossing each other is?
51.
A $120 \text{ m}$ long train is moving towards west with a speed of $10 \text{ m/s}$. A bird flying towards east with a speed of $5 \text{ m/s}$ crosses the train. The time taken by the bird to cross the train will be?
52.
Two cars A and B are moving in the same direction with velocities $30 \text{ m/s}$ and $20 \text{ m/s}$. When car A is $240 \text{ m}$ behind car B, the driver of car A applies the brakes, producing a uniform retardation of $2 \text{ m/s}^2$. Will they collide? If yes, find the time of collision. If no, find the minimum distance between them.
53.
A bus is moving with a speed of $10 \text{ m/s}$ on a straight road. A scooterist wishes to overtake the bus in $100 \text{ s}$. If the bus is at a distance of $1 \text{ km}$ from the scooterist, with what speed should the scooterist chase the bus?
Section G: Graphical Analysis of Motion
54.
What does the area under a velocity-time graph represent? What does the slope of a position-time graph represent?
55.
The displacement-time graph of a moving particle is a parabola. What can you infer about the nature of its acceleration?
56.
A particle is thrown upwards, then falls back to earth. Draw the $(a)$ position-time, $(b)$ velocity-time, and $(c)$ acceleration-time graphs for this motion. (Take upward direction as positive).
57.
A velocity-time graph of an object forms a triangle with the time axis. The base of the triangle is $10 \text{ s}$ and the peak height is $20 \text{ m/s}$. Find the total distance travelled.
58.
If the $x\text{-}t$ graph of a particle is parallel to the time axis, what is the velocity of the particle?
59.
Can a position-time graph have a negative slope? What does it signify? Can it be a vertical straight line? Give reasons.
60.
From a given $v\text{-}t$ graph which consists of a semi-circle above the $t$-axis of radius $2 \text{ m/s}$ from $t=0$ to $t=4 \text{ s}$, find the total displacement of the body.
Answer Key
Q1: $\pi : 2$
Q2: $2R$
Q3: $7\text{ m}, 5\text{ m}$
Q4: $37\text{ s}$
Q5: Yes (circle); No
Q6: (d) $\le 1$
Q7: $\sqrt{\pi^2+4}$ m
Q8: $37.5\text{ km/h}$
Q9: $0 ; 13.33\text{ m/s}$
Q10: $\frac{3uvw}{vw+uw+uv}$
Q11: $\frac{v_1 t_1 + v_2 t_2}{t_1 + t_2}$
Q12: $1.875\text{ km/h}$
Q13: $\frac{v_1 + v_2}{2}$
Q14: $9\text{ m/s}$
Q15: Yes (UCM); No in 1D
Q16: $0 ; 10\text{ m/s}$
Q17: $25\text{m}, -15\text{m/s}, 8\text{m/s}^2$
Q18: $18\text{ m/s}$
Q19: $-9\text{ m/s}$
Q20: $[LT^{-1}], [LT^{-2}], [LT^{-3}]$
Q21: $\frac{\alpha^2}{2}$
Q22: $-2av^3$
Q23: $4\text{ m}$ (distance)
Q24: $\sqrt{v_0^2 - kx^2}$
Q25: $18\text{ m/s}^2$
Q26: $1837.5\text{ m}, 105\text{ m/s}$
Q27: $21.75\text{ m}$
Q28: $-5 \times 10^5 \text{ m/s}^2$
Q29: $y = 3x$
Q30: $\frac{\alpha\beta t}{\alpha+\beta}$
Q31: Derivation
Q32: $6\text{ m/s}, 4\text{ m/s}^2$
Q33: $n^2 d$
Q34: $\sqrt{\frac{u^2+v^2}{2}}$
Q35: $8\text{ m}$
Q36: $20\text{ m}$ (from throw)
Q37: $5\text{ s}$
Q38: $41.4\%$ increase
Q39: Derivation
Q40: $\sqrt{h_1} : \sqrt{h_2}$
Q41: $3.75\text{ m}$
Q42: $4\text{ s}, 29.4\text{ m/s}$ down
Q43: $20\text{ m/s}, 15\text{m}, 20\text{m}, 15\text{m}$
Q44: $125\text{ m}$
Q45: No, $a = g$ downwards
Q46: $735\text{ m}$
Q47: $40\text{ m/s}$ South
Q48: $10\text{ m/s}$ North
Q49: $105\text{ m/s}$
Q50: $4\text{ s}$
Q51: $8\text{ s}$
Q52: No, $215\text{ m}$ apart
Q53: $20\text{ m/s}$
Q54: Displacement; Velocity
Q55: Constant Accel.
Q56: Graphs
Q57: $100\text{ m}$
Q58: Zero
Q59: Yes(-vel); No(Infinite speed)
Q60: $2\pi\text{ m}$
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