Projectile Motion

Book Subtopics

• Projectile launched horizontally
• Projectile launched at an angle
• Maximum height
• Time of flight
• Time of summit
• Horizontal range
• Maximum range
• Complementary angles producing equal ranges
• Effect of air resistance

1. Learning Outcomes

After studying this topic, students should be able to:

• Define projectile motion and trajectory.
• Explain why projectile motion is two-dimensional.
• Separate projectile motion into horizontal and vertical components.
• Analyse a horizontally launched projectile.
• Analyse a projectile launched at an angle.
• Calculate instantaneous velocity and direction.
• Derive maximum height, time of flight and range.
• Explain the condition for maximum range.
• Identify complementary angles that produce equal ranges.
• Explain the effect of air resistance.

2. Definition

3. Important Terms

Projectile

An object thrown into the air that subsequently moves under gravity.

Trajectory

The curved path followed by a projectile.

Time of Flight

The total time for which a projectile remains in the air.

Maximum Height

The greatest vertical distance reached above the point of projection.

Time of Summit

The time taken to reach maximum height.

Horizontal Range

The horizontal distance between the point of projection and the point of landing.

4. Examples of Projectile Motion

• A kicked football
• A thrown cricket ball
• A stone projected into the air
• A ball launched from a height
• Water leaving a horizontal pipe
• An object projected from the ground at an angle

5. Assumptions

The standard equations assume that:

• Air resistance is neglected.
• Gravity is the only force acting after projection.
• Acceleration due to gravity is constant.
• Earth’s rotation is neglected.
• The surface of Earth is treated as flat over the travelled distance.

6. Horizontal and Vertical Motions

Projectile motion consists of two independent motions.

Horizontal Motion

There is no horizontal acceleration: ax = 0 Therefore, horizontal velocity remains constant: vx = v_ix

Vertical Motion

The projectile accelerates downward because of gravity: a_y = −g when upward is taken as positive. Therefore, vertical velocity changes continuously.

7. Generated Diagram Placement

Projectile Motion

The formulas must also remain available as searchable HTML text and should not exist only inside the image.

33. Formula-Selection Table

Formula-Selection Table

34. Important Facts

• Projectile motion is two-dimensional.
• Horizontal and vertical motions are independent.
• Horizontal acceleration is zero without air resistance.
• Vertical acceleration is g downward.
• Horizontal velocity remains constant.
• Vertical velocity changes continuously.
• At maximum height, only the vertical velocity is zero.
• Maximum range occurs at 45° under ideal equal-level conditions.
• Complementary angles produce equal ranges for the same initial speed.
• Horizontal range formulas change if the launch and landing heights differ.

35. Key Logic

Always divide projectile motion into two independent parts: Horizontal Direction ax = 0 Use constant-velocity relations. Vertical Direction ay = −g Use uniformly accelerated motion equations. Both motions use the same value of time.

36. Tips and Tricks

• Resolve the initial velocity before starting.
• Keep horizontal and vertical calculations separate.
• Use the same time in both directions.
• At maximum height, set v_y = 0, not total v = 0.
• Use v_iy = 0 for a horizontal launch.
• Check whether landing and projection heights are equal.
• Use 45° for maximum range only under ideal equal-level conditions.
• Keep the calculator in degree mode.
• Write one mathematical step per line.

37. Common Mistakes

• Treating total velocity as zero at maximum height.
• Adding horizontal acceleration when air resistance is neglected.
• Using total initial velocity instead of its vertical component in the height formula.
• Applying equal-level range formulas to a projectile launched from a cliff.
• Using g = 0 at maximum height.
• Mixing upward-positive and downward-positive conventions.
• Using sin θ for the horizontal component.
• Forgetting the square on v_i in height and range formulas.
• Assuming air resistance keeps the trajectory parabolic.
• Confusing time of summit with total time of flight.

38. Remember Box

39. Exam Facts

• The ideal trajectory is parabolic.
• Range is maximum at 45°.
• Complementary angles produce equal ranges.
• Time of flight is twice the time of summit.
• The vertical component determines maximum height.
• The horizontal component determines horizontal motion.
• Gravity affects only the vertical component in the ideal model.
• The standard T and R formulas assume equal projection and landing heights.

40. Short-Question Preparation

Short-Question Preparation

44. Sticky Notes

Sticky Notes

• Projectile motion has horizontal and vertical components.
• a_x = 0 in the ideal model.
• a_y = −g when upward is positive.
• Horizontal velocity stays constant.
• At maximum height, v_y = 0.
• Maximum range occurs at 45°.
• Complementary angles give equal ranges.
• Standard range formulas require equal launch and landing heights.

45. 60-Second Revision

v_ix = v_i cos θ v_iy = v_i sin θ H = v_i² sin² θ2g T = 2 v_i sin θg R = v_i² sin 2θ / g R_max = v_fraci²g

46. Topic Summary

Projectile motion combines constant horizontal motion with uniformly accelerated vertical motion. Gravity changes only the vertical velocity in the ideal model. Resolving the initial velocity makes it possible to determine the projectile’s position, velocity, maximum height, time of flight and horizontal range.