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Q2.6(b):
A player throws a ball upwards with an initial speed of 29.4 m s(^{-1}).
(b) What are the velocity and acceleration of the ball at the highest point of its motion ?
Solution :
Given Variables & Initial Setup
The motion of the ball is governed by the equations of motion under constant gravitational acceleration, assuming air resistance is negligible. The following variables are identified:
- Initial velocity ($u$): $29.4\text{ m s}^{-1}$ (upwards)
- Acceleration due to gravity ($g$): $-9.8\text{ m s}^{-2}$ (downwards)
- Final velocity ($v$): Unknown at the highest point
- Acceleration ($a$): Unknown at the highest point
Step 1: Determining Velocity at the Highest Point
At the highest point of a projectile's trajectory, the vertical component of the velocity reaches a momentary state of rest before reversing direction [Per Kinematic Principles].
Mathematically, we use the first equation of motion:
$v = u + at$
At the peak, the displacement is maximum, and $v$ must be zero. If $v$ were not zero, the ball would continue to rise or have already begun to descend. Thus, at the highest point:
$v = 0\text{ m s}^{-1}$
Step 2: Determining Acceleration at the Highest Point
Acceleration is the rate of change of velocity ($a = \frac{dv}{dt}$). Even though the velocity of the ball is zero at the highest point, the gravitational force acting on the object remains constant throughout its flight [Per Newton’s Second Law of Motion].
The acceleration of an object in free fall is independent of its velocity. Therefore, at any point in the trajectory, including the highest point:
$a = g = -9.8\text{ m s}^{-2}$
The negative sign indicates that the acceleration is directed vertically downwards towards the center of the Earth.
Visual Representation of Kinematic State
Summary of Analytical Results
| Physical Quantity | Value at Highest Point |
|---|---|
| Velocity ($v$) | $0\text{ m s}^{-1}$ |
| Acceleration ($a$) | $9.8\text{ m s}^{-2}$ (downwards) |
Final Solution: At the highest point of its motion, the velocity of the ball is $0\text{ m s}^{-1}$ and its acceleration is $9.8\text{ m s}^{-2}$ directed vertically downwards.
More Questions from Class 11 Physics Motion in a Straight Line EXERCISES
- Q2.1(a): In which of the following examples of motion, can the body be considered approximately a point object: (a) a railway carriage moving without jerks between two stations.
- Q2.1(b): In which of the following examples of motion, can the body be considered approximately a point object: (b) a monkey sitting on top of a man cycling smoothly on a circular track.
- Q2.1(c): In which of the following examples of motion, can the body be considered approximately a point object: (c) a spinning cricket ball that turns sharply on hitting the ground.
- Q2.1(d): In which of the following examples of motion, can the body be considered approximately a point object: (d) a tumbling beaker that has slipped off the edge of a table.
- Q2.10(a)(i): A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h(^{-1}). Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km h(^{-1}). What is the (a) magnitude of average velocity over the interval of time (i) 0 to 30 min ?
- Q2.10(a)(ii): A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h(^{-1}). Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km h(^{-1}). What is the (a) magnitude of average velocity over the interval of time (ii) 0 to 50 min ?
- Q2.10(a)(iii): A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h(^{-1}). Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km h(^{-1}). What is the (a) magnitude of average velocity over the interval of time (iii) 0 to 40 min ?
- Q2.10(b)(i): A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h(^{-1}). Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km h(^{-1}). What is the (b) average speed of the man over the interval of time (i) 0 to 30 min ?
- Q2.10(b)(ii): A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h(^{-1}). Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km h(^{-1}). What is the (b) average speed of the man over the interval of time (ii) 0 to 50 min ?
- Q2.10(b)(iii): A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h(^{-1}). Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km h(^{-1}). What is the (b) average speed of the man over the interval of time (iii) 0 to 40 min ?
- Q2.11: We have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?
- Q2.12: Look at the graphs (a) to (d) (Fig. 2.10) carefully and state, with reasons, which of these cannot possibly represent one-dimensional motion of a particle.
- Q2.13: Figure 2.11 shows the (x-t) plot of one-dimensional motion of a particle. Is it correct to say from the graph that the particle moves in a straight line for (t) < 0 and on a parabolic path for (t) > 0 ? If not, suggest a suitable physical context for this graph.
- Q2.14: A police van moving on a highway with a speed of 30 km h(^{-1}) fires a bullet at a thief’s car speeding away in the same direction with a speed of 192 km h(^{-1}). If the muzzle speed of the bullet is 150 m s(^{-1}), with what speed does the bullet hit the thief’s car ? (Note: Obtain that speed which is relevant for damaging the thief’s car).
- Q2.15: Suggest a suitable physical situation for each of the following graphs (Fig 2.12):
- Q2.16: Figure 2.13 gives the (x-t) plot of a particle executing one-dimensional simple harmonic motion. (You will learn about this motion in more detail in Chapter13). Give the signs of position, velocity and acceleration variables of the particle at (t) = 0.3 s, 1.2 s, – 1.2 s
- Q2.17: Figure 2.14 gives the (x-t) plot of a particle in one-dimensional motion. Three different equal intervals of time are shown. In which interval is the average speed greatest, and in which is it the least ? Give the sign of average velocity for each interval.
- Q2.18: Figure 2.15 gives a speed-time graph of a particle in motion along a constant direction. Three equal intervals of time are shown. In which interval is the average acceleration greatest in magnitude? In which interval is the average speed greatest ? Choosing the positive direction as the constant direction of motion, give the signs of (v) and (a) in the three intervals. What are the accelerations at the points A, B, C and D ?
- Q2.2(a): The position-time ((x-t)) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 2.9. Choose the correct entries in the brackets below ; (a) (A/B) lives closer to the school than (B/A)
- Q2.2(b): The position-time ((x-t)) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 2.9. Choose the correct entries in the brackets below ; (b) (A/B) starts from the school earlier than (B/A)
- Q2.2(c): The position-time ((x-t)) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 2.9. Choose the correct entries in the brackets below ; (c) (A/B) walks faster than (B/A)
- Q2.2(d): The position-time ((x-t)) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 2.9. Choose the correct entries in the brackets below ; (d) A and B reach home at the (same/different) time
- Q2.2(e): The position-time ((x-t)) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 2.9. Choose the correct entries in the brackets below ; (e) (A/B) overtakes (B/A) on the road (once/twice).
- Q2.3: A woman starts from her home at 9.00 am, walks with a speed of 5 km h(^{-1}) on a straight road up to her office 2.5 km away, stays at the office up to 5.00 pm, and returns home by an auto with a speed of 25 km h(^{-1}). Choose suitable scales and plot the (x-t) graph of her motion.
- Q2.4: A drunkard walking in a narrow lane takes 5 steps forward and 3 steps backward, followed again by 5 steps forward and 3 steps backward, and so on. Each step is 1 m long and requires 1 s. Plot the (x-t) graph of his motion. Determine graphically and otherwise how long the drunkard takes to fall in a pit 13 m away from the start.
- Q2.5: A car moving along a straight highway with speed of 126 km h(^{-1}) is brought to a stop within a distance of 200 m. What is the retardation of the car (assumed uniform), and how long does it take for the car to stop ?
- Q2.6(a): A player throws a ball upwards with an initial speed of 29.4 m s(^{-1}). (a) What is the direction of acceleration during the upward motion of the ball ?
- Q2.6(c): A player throws a ball upwards with an initial speed of 29.4 m s(^{-1}). (c) Choose the (x) = 0 m and (t) = 0 s to be the location and time of the ball at its highest point, vertically downward direction to be the positive direction of (x)-axis, and give the signs of position, velocity and acceleration of the ball during its upward, and downward motion.
- Q2.6(d): A player throws a ball upwards with an initial speed of 29.4 m s(^{-1}). (d) To what height does the ball rise and after how long does the ball return to the player’s hands ? (Take (g) = 9.8 m s(^{-2}) and neglect air resistance).
- Q2.7(a): Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (a) with zero speed at an instant may have non-zero acceleration at that instant
- Q2.7(b): Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (b) with zero speed may have non-zero velocity,
- Q2.7(c): Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (c) with constant speed must have zero acceleration,
- Q2.7(d): Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (d) with positive value of acceleration must be speeding up.
- Q2.8: A ball is dropped from a height of 90 m on a floor. At each collision with the floor, the ball loses one tenth of its speed. Plot the speed-time graph of its motion between (t) = 0 to 12 s.
- Q2.9(a): Explain clearly, with examples, the distinction between : (a) magnitude of displacement (sometimes called distance) over an interval of time, and the total length of path covered by a particle over the same interval; Show that the second quantity is either greater than or equal to the first. When is the equality sign true ? [For simplicity, consider one-dimensional motion only].
- Q2.9(b): Explain clearly, with examples, the distinction between : (b) magnitude of average velocity over an interval of time, and the average speed over the same interval. [Average speed of a particle over an interval of time is defined as the total path length divided by the time interval]. Show that the second quantity is either greater than or equal to the first. When is the equality sign true ? [For simplicity, consider one-dimensional motion only].
- Q3.1(a): In which of the following examples of motion, can the body be considered approximately a point object: (a) a railway carriage moving without jerks between two stations.
- Q3.1(b): In which of the following examples of motion, can the body be considered approximately a point object: (b) a monkey sitting on top of a man cycling smoothly on a circular track.
- Q3.1(c): In which of the following examples of motion, can the body be considered approximately a point object: (c) a spinning cricket ball that turns sharply on hitting the ground.
- Q3.1(d): In which of the following examples of motion, can the body be considered approximately a point object: (d) a tumbling beaker that has slipped off the edge of a table.
- Q3.10(a): A player throws a ball upwards with an initial speed of $29.4 m s^{-1}$. (a) What is the direction of acceleration during the upward motion of the ball ?
- Q3.10(b): A player throws a ball upwards with an initial speed of $29.4 m s^{-1}$. (b) What are the velocity and acceleration of the ball at the highest point of its motion ?
- Q3.10(c): A player throws a ball upwards with an initial speed of $29.4 m s^{-1}$. (c) Choose the $x = 0 m$ and $t = 0 s$ to be the location and time of the ball at its highest point, vertically downward direction to be the positive direction of $x$-axis, and give the signs of position, velocity and acceleration of the ball during its upward, and downward motion.
- Q3.10(d): A player throws a ball upwards with an initial speed of $29.4 m s^{-1}$. (d) To what height does the ball rise and after how long does the ball return to the player’s hands ? (Take $g = 9.8 m s^{-2}$ and neglect air resistance).
- Q3.11(a): Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (a) with zero speed at an instant may have non-zero acceleration at that instant
- Q3.11(b): Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (b) with zero speed may have non-zero velocity,
- Q3.11(c): Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (c) with constant speed must have zero acceleration,
- Q3.11(d): Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion (d) with positive value of acceleration must be speeding up.
- Q3.12: A ball is dropped from a height of $90 m$ on a floor. At each collision with the floor, the ball loses one tenth of its speed. Plot the speed-time graph of its motion between $t = 0$ to $12 s$.
- Q3.13(a): Explain clearly, with examples, the distinction between : (a) magnitude of displacement (sometimes called distance) over an interval of time, and the total length of path covered by a particle over the same interval;
- Q3.13(b): Explain clearly, with examples, the distinction between : (b) magnitude of average velocity over an interval of time, and the average speed over the same interval. [Average speed of a particle over an interval of time is defined as the total path length divided by the time interval]. Show in both (a) and (b) that the second quantity is either greater than or equal to the first. When is the equality sign true ? [For simplicity, consider one-dimensional motion only].
- Q3.14(a)(i): A man walks on a straight road from his home to a market $2.5 km$ away with a speed of $5 km h^{-1}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 km h^{-1}$. What is the (a) magnitude of average velocity of the man over the interval of time (i) 0 to 30 min
- Q3.14(a)(ii): A man walks on a straight road from his home to a market $2.5 km$ away with a speed of $5 km h^{-1}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 km h^{-1}$. What is the (a) magnitude of average velocity of the man over the interval of time (ii) 0 to 50 min
- Q3.14(a)(iii): A man walks on a straight road from his home to a market $2.5 km$ away with a speed of $5 km h^{-1}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 km h^{-1}$. What is the (a) magnitude of average velocity of the man over the interval of time (iii) 0 to 40 min ?
- Q3.14(b)(i): A man walks on a straight road from his home to a market $2.5 km$ away with a speed of $5 km h^{-1}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 km h^{-1}$. What is the (b) average speed of the man over the interval of time (i) 0 to 30 min
- Q3.14(b)(ii): A man walks on a straight road from his home to a market $2.5 km$ away with a speed of $5 km h^{-1}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 km h^{-1}$. What is the (b) average speed of the man over the interval of time (ii) 0 to 50 min
- Q3.14(b)(iii): A man walks on a straight road from his home to a market $2.5 km$ away with a speed of $5 km h^{-1}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 km h^{-1}$. What is the (b) average speed of the man over the interval of time (iii) 0 to 40 min ? [Note: You will appreciate from this exercise why it is better to define average speed as total path length divided by time, and not as magnitude of average velocity. You would not like to tell the tired man on his return home that his average speed was zero !]
- Q3.15: When we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why ?
- Q3.16: Look at the graphs (a) to (d) (Fig. 3.20) carefully and state, with reasons, which of these cannot possibly represent one-dimensional motion of a particle.
- Q3.17: Figure 3.21 shows the $x-t$ plot of one-dimensional motion of a particle. Is it correct to say from the graph that the particle moves in a straight line for $t < 0$ and on a parabolic path for $t > 0$ ? If not, suggest a suitable physical context for this graph.
- Q3.18: A police van moving on a highway with a speed of $30 km h^{-1}$ fires a bullet at a thief’s car speeding away in the same direction with a speed of $192 km h^{-1}$. If the muzzle speed of the bullet is $150 m s^{-1}$, with what speed does the bullet hit the thief’s car ? (Note: Obtain that speed which is relevant for damaging the thief’s car).
- Q3.19: Suggest a suitable physical situation for each of the following graphs (Fig 3.22):
- Q3.2(a): The position-time ($x-t$) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 3.19. Choose the correct entries in the brackets below ; (a) (A/B) lives closer to the school than (B/A)
- Q3.2(b): The position-time ($x-t$) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 3.19. Choose the correct entries in the brackets below ; (b) (A/B) starts from the school earlier than (B/A)
- Q3.2(c): The position-time ($x-t$) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 3.19. Choose the correct entries in the brackets below ; (c) (A/B) walks faster than (B/A)
- Q3.2(d): The position-time ($x-t$) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 3.19. Choose the correct entries in the brackets below ; (d) A and B reach home at the (same/different) time
- Q3.2(e): The position-time ($x-t$) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. 3.19. Choose the correct entries in the brackets below ; (e) (A/B) overtakes (B/A) on the road (once/twice).
- Q3.20: Figure 3.23 gives the $x-t$ plot of a particle executing one-dimensional simple harmonic motion. (You will learn about this motion in more detail in Chapter14). Give the signs of position, velocity and acceleration variables of the particle at $t = 0.3 s$, $1.2 s$, $– 1.2 s$.
- Q3.21: Figure 3.24 gives the $x-t$ plot of a particle in one-dimensional motion. Three different equal intervals of time are shown. In which interval is the average speed greatest, and in which is it the least ? Give the sign of average velocity for each interval.
- Q3.22: Figure 3.25 gives a speed-time graph of a particle in motion along a constant direction. Three equal intervals of time are shown. In which interval is the average acceleration greatest in magnitude ? In which interval is the average speed greatest ? Choosing the positive direction as the constant direction of motion, give the signs of $v$ and $a$ in the three intervals. What are the accelerations at the points $A$, $B$, $C$ and $D$ ?
- Q3.3: A woman starts from her home at 9.00 am, walks with a speed of $5 km h^{-1}$ on a straight road up to her office $2.5 km$ away, stays at the office up to 5.00 pm, and returns home by an auto with a speed of $25 km h^{-1}$. Choose suitable scales and plot the $x-t$ graph of her motion.
- Q3.4: A drunkard walking in a narrow lane takes 5 steps forward and 3 steps backward, followed again by 5 steps forward and 3 steps backward, and so on. Each step is $1 m$ long and requires $1 s$. Plot the $x-t$ graph of his motion. Determine graphically and otherwise how long the drunkard takes to fall in a pit $13 m$ away from the start.
- Q3.5: A jet airplane travelling at the speed of $500 km h^{-1}$ ejects its products of combustion at the speed of $1500 km h^{-1}$ relative to the jet plane. What is the speed of the latter with respect to an observer on the ground ?
- Q3.6: A car moving along a straight highway with speed of $126 km h^{-1}$ is brought to a stop within a distance of $200 m$. What is the retardation of the car (assumed uniform), and how long does it take for the car to stop ?
- Q3.7: Two trains $A$ and $B$ of length $400 m$ each are moving on two parallel tracks with a uniform speed of $72 km h^{-1}$ in the same direction, with $A$ ahead of $B$. The driver of $B$ decides to overtake $A$ and accelerates by $1 m s^{-2}$. If after $50 s$, the guard of $B$ just brushes past the driver of $A$, what was the original distance between them ?
- Q3.8: On a two-lane road, car A is travelling with a speed of $36 km h^{-1}$. Two cars $B$ and $C$ approach car $A$ in opposite directions with a speed of $54 km h^{-1}$ each. At a certain instant, when the distance $AB$ is equal to $AC$, both being $1 km$, $B$ decides to overtake $A$ before $C$ does. What minimum acceleration of car $B$ is required to avoid an accident ?
- Q3.9: Two towns $A$ and $B$ are connected by a regular bus service with a bus leaving in either direction every $T$ minutes. A man cycling with a speed of $20 km h^{-1}$ in the direction $A$ to $B$ notices that a bus goes past him every $18 min$ in the direction of his motion, and every $6 min$ in the opposite direction. What is the period $T$ of the bus service and with what speed (assumed constant) do the buses ply on the road?
CBSE Solutions for Class 11 Physics Motion in a Straight Line
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