Calculating Time to Maximum Height in Vertical Motion
This question asks us to determine the time it takes for a ball, thrown vertically upward with a specific initial speed, to reach its maximum height. This scenario is a classic example of motion under constant acceleration due to gravity.
Understanding Vertical Projectile Motion
When an object is thrown vertically upward, it experiences a downward acceleration due to gravity. This acceleration causes the object's upward velocity to decrease continuously until it momentarily becomes zero at the highest point of its trajectory. After reaching the maximum height, the object starts falling back down.
Key points for motion to maximum height:
- The initial velocity is upward.
- The acceleration is downward (due to gravity), taken as negative if upward is positive.
- At the maximum height, the instantaneous velocity of the object is zero.
Identifying Given Information and Goal
- Initial velocity of the ball (\(u\)) = 40 m/s (upward)
- Acceleration due to gravity (\(g\)) = 9.8 m/s² (downward)
- Final velocity at maximum height (\(v\)) = 0 m/s
- We need to find the time taken (\(t\)) to reach this maximum height.
Since we are considering motion under constant acceleration, we can use the equations of kinematics. We need an equation that relates initial velocity, final velocity, acceleration, and time.
Applying the Correct Kinematic Equation
The appropriate kinematic equation is:
\(v = u + at\)
Here, \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time.
In our case, the acceleration \(a\) is the acceleration due to gravity, \(g\). Since the initial velocity is upward and gravity acts downward, we take \(u\) as positive and \(a = -g\).
So the equation becomes:
\(v = u - gt\)
Step-by-Step Calculation
We are given:
- \(u = 40\) m/s
- \(v = 0\) m/s (at maximum height)
- \(g = 9.8\) m/s²
Substitute these values into the equation \(v = u - gt\):
\(0 = 40 - (9.8) \times t\)
Now, we solve for \(t\):
\(9.8t = 40\)
\(t = \frac{40}{9.8}\)
Calculating the value of \(t\):
\(t \approx 4.0816\) seconds
Comparing with the Options
The calculated time is approximately 4.08 seconds. Let's look at the given options:
- 2 s
- 3 s
- 4 s
- 5 s
The value 4.08 seconds is closest to 4 seconds among the given options.
| Quantity | Symbol | Value |
|---|---|---|
| Initial Velocity | \(u\) | 40 m/s |
| Final Velocity (at max height) | \(v\) | 0 m/s |
| Acceleration (gravity) | \(a\) | -9.8 m/s² |
| Time | \(t\) | ? |
Conclusion on Time to Reach Maximum Height
Using the kinematic equation \(v = u + at\), and considering the final velocity is zero at the maximum height, we calculated the time taken for the ball to reach the maximum height to be approximately 4.08 seconds. Comparing this value with the provided options, 4 seconds is the closest approximation.
Revision Table: Key Concepts
| Concept | Description | Relevance to Problem |
|---|---|---|
| Vertical Projectile Motion | Motion of an object thrown upward or downward under the influence of gravity only. | The ball's movement is an example of this. |
| Acceleration due to Gravity (\(g\)) | Constant acceleration experienced by objects near the Earth's surface, approximately 9.8 m/s² downward. | This causes the velocity change of the ball. |
| Maximum Height | The highest point reached by a projectile, where its instantaneous vertical velocity is zero. | This point is the target for finding the time \(t\). |
| Kinematic Equations | Equations relating displacement, initial velocity, final velocity, acceleration, and time for motion with constant acceleration. | \(v = u + at\) was used to solve the problem. |
Additional Information: Motion Symmetries and Gravity
Understanding vertical motion under gravity involves a few important points:
- Constant Acceleration: The acceleration due to gravity is considered constant near the Earth's surface (ignoring air resistance). This makes the kinematic equations applicable.
- Direction Matters: It is crucial to define a positive direction (e.g., upward) and consistently use signs for velocity, displacement, and acceleration. Since gravity acts downward, its acceleration is negative if upward is positive.
- Time Symmetry: In the absence of air resistance, the time taken for the object to reach its maximum height from the point of projection is equal to the time taken for it to fall back from the maximum height to the point of projection.
- Speed Symmetry: The speed of the object at any given height above the point of projection while going up is equal to its speed at the same height while coming down.
These principles help analyze the entire trajectory of a vertically projected object.
Important Questions from Kinematic equations for uniformly accelerated motion
A tennis ball is thrown in the vertically upward direction and the ball attains a maximum height of 20 m. The ball was thrown approximately with an upward velocity of
A particle experiences constant acceleration for 20 s after starting from rest. If it travels a distance X 1, in the first 10 s and distance X 2in the remaining 10 s, then which of the following is true?
- A ball thrown up vertically returns to the ground after 10 second. Find the velocity with which it was thrown up? (if g = 10 m/s2).
Which of the following best describes the relationship between distance, time, and speed when a body is NOT accelerating?
If the distance travelled by a body in the $n^{th}$ second is given by $(7 + 5n)$ m, then find the initial velocity and acceleration of the body respectively.
