PROJECTILE MOTION Lesson Note
LEARNING OBJECTIVES
This SS2 Physics lesson note covers PROJECTILE MOTION for First Term, Week 4.
Lesson Note
LEARNING OBJECTIVES
By end of lesson, students should be able to:
1. Define projectile and projectile motion
2. Give examples of projectiles
3. Explain types of projectile motion
4. Explain terms: Trajectory, Time of flight, Range, Maximum height
5. Derive formulas for projectile
6. Solve problems on projectile
A. DEFINITION
A projectile is anybody/object that is thrown or projected with an initial velocity and then allowed to move freely under gravity alone, without self-propulsion.
Projectile Motion is the motion of a projectile under gravity.
Key point: Once projected, the only force acting is gravity (g = 10 m/s² downwards). Air resistance is ignored.
Examples: A thrown ball, a fired bullet, a kicked football, a bomb dropped from an aeroplane, a stone thrown up, a javelin.
B. TYPES OF PROJECTILE MOTION
a) Horizontal Projection: Body projected horizontally from a height. Example: Ball rolling off a table, bomb dropped from plane.
b) Projectile Fired at an Angle to Horizontal: Body thrown up at an angle (most common). Example: Football kicked, bullet fired at angle. This is also called Parabolic Projectile.
C. IMPORTANT CONCEPTS
The motion of a projectile is in 2 dimensions:
- Horizontal (x-axis): Velocity is CONSTANT (ax = 0) because no gravity acts horizontally.
Ux = U Cosθ
- Vertical (y-axis): Velocity CHANGES because gravity acts. (ay = g = 10 m/s²)
Uy = U Sinθ
The path followed by a projectile is called TRAJECTORY and it is parabolic in shape.
D. TERMS IN PROJECTILE MOTION (For projectile fired at angle θ)
Let U = initial velocity, θ = angle of projection, g = 10 m/s²
i) Time of Flight (T): Total time the projectile stays in air.
T = 2U Sinθ / g
ii) Maximum Height (H): Highest vertical distance reached. At this point vertical velocity = 0.
H = (U Sinθ)² / 2g = U² Sin²θ / 2g
iii) Range (R): Horizontal distance covered from point of projection to point where it lands (same level).
R = U² Sin2θ / g
Note: Sin2θ = 2 Sinθ Cosθ
For maximum range, θ = 45°, Sin90° = 1, so Rmax = U² / g
iv) Time to reach maximum height (t_up): = U Sinθ / g = T/2
PROJECTILE PROJECTED HORIZONTALLY
If a body is projected horizontally with velocity U from height h:
- Initial vertical velocity = 0
- Time to reach ground: From h = ½gt² => t = √(2h/g)
- Range (horizontal distance): R = U _ t = U√(2h/g)
- Vertical velocity on reaching ground: v = gt
WORKED EXAMPLES
Example 1: A stone is projected with velocity 20 m/s at 30° to horizontal. Calculate:
a) Time of flight b) Max height c) Range (g=10)
Solution: U=20, θ=30°, Sin30=0.5, Sin60=0.866
a) T = 2U Sinθ / g = 2_20_0.5 /10 = 20/10 = 2s
b) H = U² Sin²θ /2g = 20² _ (0.5)² /20 = 400_0.25/20 = 5m
c) R = U² Sin2θ /g = 400 _ Sin60 /10 = 400_0.866/10 = 34.64m
Example 2: A bomb is dropped from plane flying horizontally at 100 m/s at height 500m. Find where it lands (g=10).
t = √(2h/g) = √(1000/10) = √100 =10s
Range = U _ t = 100 10 = *1000m from point below drop point.
E. APPLICATIONS OF PROJECTILES
1. In sports: football, basketball, javelin, long jump
2. In warfare: firing of gun, cannon, launching missiles
3. In engineering
Evaluation
OBJECTIVE QUESTIONS
1. A body thrown up and allowed to move under gravity only is called:
A. Projected body
B. Projectile
C. Free body
D. Missile
2. The path followed by a projectile is called:
A. Orbit
B. Trajectory
C. Parabola
D. Curve
3. The shape of trajectory of a projectile is:
A. Circular
B. Straight line
C. Parabolic
D. Hyperbolic
4. For a projectile, the horizontal component of velocity is:
A. Increasing
B. Decreasing
C. Constant
D. Zero
5. At maximum height, the vertical velocity of a projectile is:
A. Maximum
B. Zero
C. Equal to U
D. Double
6. The time of flight T for projectile at angle θ is:
A. U Sinθ / g
B. 2U Sinθ / g
C. U² Sinθ / g
D. U Cosθ / g
7. Maximum height H is given by:
A. U² Sin²θ / 2g
B. U² Sin2θ / g
C. U Cosθ / g
D. 2U Sinθ / g
8. For maximum horizontal range, angle of projection should be:
A. 30°
B. 45°
C. 60°
D. 90°
9. A bomb dropped from an aeroplane moving horizontally is an example of:
A. Vertical projection
B. Horizontal projection
C. Angular projection
D. Zero projection
10. The range of a horizontal projectile from height h is:
A. U * t
B. U√(2h/g)
C. Both A and B
D. h/g
THEORY QUESTIONS
1. a) Define projectile motion and trajectory.
b) State two types of projectile motion with one example each.
c) A stone is projected with velocity 20 m/s at an angle 30° to the horizontal. Calculate:
i) Time of flight
ii) Maximum height
iii) Horizontal range [Take g = 10 m/s², Sin30=0.5, Sin60=0.866]
2. a) Derive expressions for Time of flight and Maximum height for a projectile fired at angle θ.
b) Explain why the horizontal velocity of a projectile remains constant.
c) A body is projected horizontally with velocity 15 m/s from the top of a tower 80m high. Calculate:
i) Time to reach ground
ii) How far from foot of tower it hits ground