Kinematics-2D

From the image given below, a lift of height $32\text{ m}$ is going up with acceleration $a = 6\text{ m/s}^2$. When the velocity of the lift is $15\text{ m/s}$ upward, a coin is dropped from the ceiling at $t = 0$. Find the time when the coin will hit the floor.

From the image given below, at $t = 0$ a coin is dropped from the ceiling of a lift of height $63\text{ m}$ which is moving upwards with an acceleration $a = 4\text{ m/s}^2$ and velocity $v = 19.7\text{ m/s}$. Find the time when the coin will hit the floor.

From the image given below, a ball is thrown downward at $10\text{ m/s}$ with respect to a lift of height $100\text{ m}$ which is moving with an acceleration $a = 2\text{ m/s}^2$ and velocity $v = 40\text{ m/s}$. When will the coin strike the floor?

From the image given below, when will the coin strike the floor of the lift, given its height is $120\text{ m}$, acceleration $a = 5\text{ m/s}^2$ upwards, and initial velocity $v = 10\text{ m/s}$ upwards?

From the image given below, when will the coin strike the floor of the lift, given its height is $120\text{ m}$, acceleration $a = 5\text{ m/s}^2$ upwards, and initial velocity $v = 10\text{ m/s}$ upwards?

A boat is sent across a river with a velocity of $8\text{ km/hr}$. If the resultant velocity of boat is $10\text{ km/hr}$, then velocity of the river is :

A boat is moving with velocity of $3\hat{i} + 4\hat{j}$ in river (wrt ground) and water is moving with a velocity of $-3\hat{i} - 4\hat{j}$ with respect to ground. Relative velocity of boat with respect to water is :

A swimmer can swim in still water with a speed of $\sqrt{5}\text{ m/s}$. While crossing a river his average speed is $3\text{ m/s}$. If he cross the river in the shortest possible time, what is the speed of flow of water?

A river is flowing from W to E with a speed of $5\text{ m/min}$. A man can swim in still water with a velocity $10\text{ m/min}$. In which direction should the man swim so as to take the shortest possible path to go to the south.

A man can swim in still water at $6\text{ km/h}$. He want to reach a point just opposite his starting point. In which direction he should swim, if the river is flowing at $3\text{ km/h}$ ? from the image given below

A man can swim in still water at $6\text{ km/h}$. He want to reach a point just opposite his starting point. In which direction he should swim, if the river is flowing at $3\text{ km/h}$ ? from the image given below Calculate the time taken by the man to cross the river at right angles to the flow; if the width of the river is $\sqrt{3}\text{ km}$.

```python
# Let's verify the problem statement and calculations from the image.
# The question in the image (inside the highlighted area - AoI) is:
# "Q. A river is flowing with velocity $5\text{ m/s}$ wrt ground. A man can swim with velocity $10\text{ m/s}$ wrt river. The width of the river is $100\text{ m}$.
# (1) In which direction man should swim so that he crosses that river in min distance travel also find time taken to cross the river."
# Wait, let's check the solution in the image:
# v_{m/r} = v = 10 m/s
# u = 5 m/s
# d = 100 m
# v \sin\theta = u => 10 \sin\theta = 5 => \sin\theta = 1/2 => \theta = 30^\circ
# Let's write code to verify the calculation.
v = 10
u = 5
d = 100
import math
theta_deg = math.degrees(math.asin(u / v))
print(f"theta: {theta_deg}")
# Minimum distance to cross is width d = 100 m.
# Time taken: t = d / (v * cos(theta))
theta_rad = math.radians(theta_deg)
t = d / (v * math.cos(theta_rad))
print(f"time: {t}")
```
```text
theta: 30.000000000000004
time: 11.547005383792516
```

From the image given below, a river is flowing with velocity $5\text{ m/s}$ wrt ground. A man can swim with velocity $10\text{ m/s}$ wrt river. The width of the river is $100\text{ m}$. In which direction man should swim so that he crosses that river in min distance travel also find time taken to cross the river.

```python
# Let's verify the calculation for the question selected in the AoI:
# "Q. In following figure if man is swimming with velocity $10\text{ m/s}$ wrt river at an angle of $37^\circ$ to the river velocity. Find time taken to cross the river and drift if velocity of river is $10\text{ m/s}$ and width of the river is $30\text{ m}$"
# Let's check the text in the image:
# v_{m/r} = 10 m/s
# angle with river velocity = 37°
# width of river d = 30 m
# river velocity u = 10 m/s
# time to cross: t = d / (v * sin(37°)) = 30 / (10 * sin(37°)) = 30 / (10 * 0.6) = 30 / 6 = 5 sec.
# component of v along river = v * cos(37°) = 10 * 0.8 = 8 m/s
# total velocity along river = u + v * cos(37°) = 10 + 8 = 18 m/s
# drift = (u + v * cos(37°)) * t = 18 * 5 = 90 m.
import math
d = 30
v = 10
u = 10
theta = math.radians(37)
t = d / (v * math.sin(theta))
v_net = u + v * math.cos(theta)
drift = v_net * t
print(f"Time: {t}")
print(f"Drift: {drift}")
```
```text
Time: 4.984920423367449
Drift: 89.66054888228679
```

From the image given below, in the following figure if a man is swimming with a velocity $10\text{ m/s}$ with respect to the river at an angle of $37^\circ$ to the river velocity, find the time taken to cross the river and the drift if the velocity of the river is $10\text{ m/s}$ and the width of the river is $30\text{ m}$.