Motion in a Gravitational Field Revision Notes for Grade 12 NSC Matric Physical Sciences

Motion in a Gravitational Field

What is vertical projectile motion?

Vertical projectile motion occurs when objects are thrown, shot or dropped directly upward, downward, or when objects fall under gravity alone. This type of motion is something we observe daily - from dropping a ball to throwing something into the air.

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A projectile is defined as any body or particle that is launched and then moves under the influence of only gravity. The key point here is that once the object is in motion, gravity is the only force acting on it (we ignore air resistance in these calculations).

Understanding motion in Earth's gravitational field

According to Newton's Law of Universal Gravitation, any object in Earth's gravitational field experiences a force that pulls it towards the centre of the Earth. This creates a constant gravitational acceleration that we represent with the symbol g.

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Key facts about gravitational acceleration:

g = 9.8 m⋅s⁻² (this value is constant for everyday problems)
Gravity always acts downward towards Earth's centre
The direction of gravity never changes, regardless of whether an object moves up or down
In reality, g varies slightly with distance from Earth, but we treat it as constant for calculations

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The diagram shows an important principle: whether an object moves upward or downward, the gravitational acceleration g always points towards Earth's centre. This means that objects moving upward experience a downward acceleration, while objects moving downward also experience the same downward acceleration.

Characteristics of vertical projectile motion

Initial velocity vs gravitational acceleration

It's crucial to understand that initial velocity ( $v_i$ ) and gravitational acceleration ( $g$ ) are completely different quantities:

Initial velocity depends on how the object was launched
Gravitational acceleration is always constant at 9.8 m⋅s⁻² downward

The journey of a projectile

When an object is thrown upward with initial velocity, it follows a predictable pattern that demonstrates the constant influence of gravity throughout its motion.

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Worked Example: Motion of a Ball Thrown Upward

When an object is thrown upward with initial velocity:

Step 1: It starts with maximum upward velocity

Step 2: Gravity gradually reduces this upward velocity

Step 3: At maximum height, the velocity becomes zero ( $v = 0$ m⋅s⁻¹)

Step 4: The object then begins falling downward

Step 5: Gravity accelerates it downward, increasing its speed

This diagram illustrates the complete motion of a ball thrown upward. Notice how:

The velocity vectors (red arrows) show the changing speed and direction
At maximum height ( $h_{max}$ ), the velocity is zero
The motion is symmetric around the maximum height point

Time symmetry in projectile motion

One of the most useful principles in vertical projectile motion is time symmetry. This fundamental concept greatly simplifies calculations and provides insight into the nature of gravitational motion.

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Time to reach maximum height = Time to fall back to original position

This symmetry has two important implications:

Time intervals: The time taken to rise from initial position to maximum height equals the time taken to fall from maximum height back to the initial position

Magnitude of velocity: The magnitude (size) of velocity at corresponding points during upward and downward motion is the same. However, the direction is opposite (upward vs downward)

This simple diagram shows how an object's motion changes over time - starting slowly spaced (slow motion) and becoming more densely packed (faster motion) as gravity accelerates the object.

Applying equations of motion

The equations of rectilinear motion from Grade 10 can be directly applied to vertical projectile motion because:

Motion occurs in a straight line (vertical direction only)
Acceleration is constant ( $g = 9.8$ m⋅s⁻²)
We can treat this as one-dimensional motion

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Important reminders for calculations:

Always choose a consistent positive direction (usually upward is positive)
Remember that $g = -9.8$ m⋅s⁻² if upward is positive (since gravity acts downward)
Use vector concepts - magnitude and direction both matter
Initial velocity and acceleration are independent quantities

Key equations to remember:

The fundamental kinematic equations for vertical projectile motion are:

Final velocity: $v = v_i + gt$
Displacement: $s = v_i t + \frac{1}{2}gt^2$
Velocity-displacement relationship: $v^2 = v_i^2 + 2gs$

Exam tips for vertical projectile motion

Understanding common pitfalls and developing a systematic approach to problem-solving will significantly improve your performance on vertical projectile motion questions.

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Common exam traps to avoid:

Don't confuse initial velocity with gravitational acceleration
Remember that at maximum height, velocity = 0, but acceleration is still g
Use time symmetry to solve complex problems more easily
Always define your positive direction clearly
Check that your final answers make physical sense

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Worked Example: Problem-solving strategy

Step 1: Identify what type of motion is occurring

Step 2: Choose your positive direction

Step 3: List known values and what you need to find

Step 4: Select the most appropriate equation

Step 5: Substitute values carefully, watching signs

Step 6: Check your answer makes sense

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Key Points to Remember:

Projectile motion involves objects moving under gravity alone, with constant acceleration $g = 9.8$ m⋅s⁻² downward
Time symmetry means the time to reach maximum height equals the time to return to the starting position
Gravitational acceleration is always constant and always points towards Earth's centre, regardless of the object's motion direction
At maximum height, velocity equals zero but acceleration is still $g$ (the object is still being pulled by gravity)
Kinematic equations from Grade 10 apply directly to vertical projectile motion since it's motion in a straight line with constant acceleration

Motion in a Gravitational Field (Grade 12 NSC Matric Physical Sciences): Revision Notes