PROJECTILE MOTION
1) A projectile is an object that is projected (launched) into the air by throwing or kicking or
firing eg ball, bullet.
2) The path followed by the projectile is called its trajectory and is determined by the effects
of gravity and air resistance.
3) If we ignore air resistance, then the only force acting on the projectile after launch is
gravitational force (or weight) which acts downwards. In this instance the path followed
by the projectile is symmetrical.
4) The projectile is therefore in freefall and experiences an acceleration of g (g = 9,8 m/s 2)
downwards throughout its motion whether it is travelling up or down.
One-dimensional projectiles (vertical projectiles) – Objects in freefall
These projectiles move vertically up and/or down hence their trajectory is a straight line. They
all have the same vertical acceleration namely g = 9.8 m.s-2 downwards.
There are 4 variations:
Object Object thrown Object thrown up Object thrown up
dropped from down with an with an initial with an initial
rest above the initial velocity velocity & returns velocity from a
ground to point of launch point above earths
surface
vi = 0 m.s-1 vi ≠ 0 m.s-1 vi ≠ 0 m.s-1 vi ≠ 0 m.s-1
g = 9.8 m.s-2 g = 9.8 m.s-2 g = 9.8 m.s-2 down g = 9.8 m.s-2 down
down down throughout trip. As it throughout the trip.
travels up, vel dec v at max height = o
uniformly until it m/s.
equals o m/s at max
height. tup = tdown
vup = vdown at same
point.
Problem Solving:
Draw a sketch.
State which trip/ part of trip you are working with.
State the SIGN CONVENTION (up as + or down as +). Use the sign convention when
assigning values to vi , vf , a and Δy (vectors). They can assume + or – values depending
on the sign convention. Δt can only be positive (scalar).
Page 1 of 6
We use Δy for displacement instead of Δx , because the motion is in vertical plane.
Example 1:
A boy standing on top of a tall building accidentally drops his ball. The ball takes 2 s to reach
the ground.
1.1) Calculate the velocity with which the ball strikes the ground.
1.2) Calculate the height of the building.
1.3) Draw sketch graphs of y vs Δt ; v vs Δt and a vs Δt using the ground as the reference
point and down as positive. On the same system of axes , use a different colour (red) to
draw the same graphs if up as positive is used.
1.4) Repeat (1.3) but this time use the point of launch as the reference point.
Example 2:
A boy standing at the top of a building 100 m high throws his ball downwards to the ground. The
ball reaches the ground with a velocity of 20 m/s.
2.1) Calculate the velocity with which the ball was thrown.
2.2) Calculate the time taken to reach the ground.
2.3) Draw sketch graphs of y vs Δt ; v vs Δt and a vs Δt using the ground as the reference
point and down as positive. On the same system of axes , use a different colour (red) to
draw the same graphs if up as positive is used.
2.4) Repeat (2.3) but this time use the point of launch as the reference point.
Example 3:
An object is fired vertically upwards from the ground level with an initial velocity of 50 m/s.
3.1) Calculate the maximum height above the ground that the object reaches.
3.2) Calculate the total time taken for the object to reach the ground. Work this out using
upward trip only, downward trip only and whole trip.
3.3) Draw graphs of y vs t, v vs t and a vs t using the convention up as positive.
3.4) Draw on the same system of axes the same graphs if downwards as positive is used.
Use a different colour.
Example 4:
A girl standing on the edge of a cliff throws a stone vertically upwards with a velocity of 5 m/s.
The stone strikes the water surface below 3 s later. Calculate:
4.1) Maximum height above the surface of the water that the stone reaches.
4.2) Height of the cliff.
4.3) Time taken for the stone to be 20 m above the surface of the water.
4.4) Displacement relative to the cliff after 1,2 s.
4.5) Time taken for the stone to reach a velocity of 2 m/s.
4.6) Draw sketch graphs of y vs Δt ; v vs Δt and a vs Δt using the water as the reference
point and down as positive. On the same system of axes , use a different colour (red) to
draw the same graphs if up as positive is used.
4.7) Repeat (4.6) but this time use the point of launch as the reference point.
Application:
1) A hot air balloon is rising at 5 m/s. When the balloon is 100 m above the ground, a
sandbag is dropped from it. Calculate:
1.1) The maximum height reached by the sandbag.
1.2) The time taken for the sandbag to reach the ground after it was released.
Page 2 of 6
1.3) The velocity with which the bag reaches the ground.
1.4) Draw the y vs Δt graph for the sandbag indicating all relevant values.
2) An elevator is rising up at a constant velocity of 15 m/s. At a certain instant, a screw at
the bottom of the elevator comes loose and falls taking 5 s to reach the ground below.
2.1) Calculate the height of the elevator above the ground when the screw breaks
loose and falls.
2.2) Calculate the height of the elevator the instant the screw touches the ground.
Assume that the elevator continues to rise at a constant velocity.
2.3) On the same system of axes, draw the v vs Δt graph for the screw and the
elevator.
3) A tennis ball is dropped from a certain height above the ground. The ball continues to
bounce until it comes to a standstill. Use the ground as the origin and take up as positive to
draw a y vs t and v vs t graph of the motion.
4) Any falling object which is being acted upon only by the force of gravity is said to be
in a state of free fall.
4.1) Briefly describe how you can make use of a small free-falling stone to determine
how deep the water level is in a well (represented by y in the diagram below).
4.2) Give ONE reason why the concept of free fall might not give a correct answer.
4.3) A student is at the top of a building of height h. He throws a stone, X, upward
with a speed v. He then throws a second identical stone, Y, downward at the
same speed v.
4.3.1 Redraw the following set of axes in the ANSWER BOOK and sketch the
graphs of position versus time for each of the stones X and Y. Use the
ground as the point of zero position.
4.3.2 How will the velocities of the two stones, X and Y, compare when they reach
the ground? Explain your answer.
Page 3 of 6
5) A mountain climber stands at the top of a 50m cliff that overhangs a calm pool. She
throws two stones vertically downward 1s apart and observes that the two stones
reach the water simultaneously after a while. The first stone was thrown at an initial
speed of 2 m.s-1.
Calculate the initial speed at which she threw the second stone. Ignore the effects of
friction.
6) A hot-air balloon is rising vertically at constant velocity. When the balloon is at a
height of 88m above the ground, a stone is released from it. The displacement-time
graph below represents the motion of the stone from the moment it is released from
the balloon until it strikes the ground. Ignore the effect of air resistance.
Use information from the graph to answer the following questions:
6.1 Calculate the velocity of the hot-air balloon at the instant the stone is released. (6)
6.2 Draw a sketch graph of velocity versus time for the motion of the stone from the
moment it is released from the balloon until it strikes the ground. Indicate the
respective values of the intercepts on your velocity-time graph. (3)
7) A supervisor, 1,8 m tall, visits a construction site. A brick resting at the edge of a roof
50 m above the ground suddenly falls. At the instant when the brick has fallen 30m
Page 4 of 6
the supervisor sees the brick coming down directly towards him from above.
Ignore the effects of friction and take the downwards motion as positive.
7.1 Calculate the speed of the brick after it has fallen 30 m. (3)
7.2 The average reaction time of a human being is 0,4s. With the aid of a suitable
calculation, determine whether the supervisor will be able to avoid being hit by the
brick. (6)
8) A man fires a projectile X vertically upwards at a velocity of 29,4 m.s-1 from the
EDGE of a cliff of height 100m. After some time the projectile lands on the ground
below the cliff. The velocity-time graph below (NOT DRAWN TO SCALE) represents
the motion of projectile X. (Ignore the effects of friction.)
8.1 Use the graph to determine the time that the projectile takes to reach its maximum
height. (A calculation is not required.) (1)
8.2 Calculate the maximum height that projectile X reaches above the ground. (4)
8.3 Sketch the position-time graph for projectile X for the period t = 0s to t = 6s.
USE THE EDGE OF THE CLIFF AS ZERO OF POSITION.
Indicate the following on the graph:
• The time when projectile X reaches its maximum height
• The time when projectile X reaches the edge of the cliff (4)
8.4 One second (1s) after projectile X is fired, the man's friend fires a second
projectile Yupwards at a velocity of 49m.s-1 FROM THE GROUND BELOW THE
CLIFF.
The first projectile, X, passes projectile Y 5,23 s after projectile X is fired.
(Ignore the effects of friction.)
Calculate the following:
8.4.1 The velocity of projectile X at the instant it passes projectile Y (5)
8.4.2 The velocity of projectile X RELATIVE to projectile Y at the instant it passes
projectile Y(5)
Page 5 of 6
9) A stone is thrown vertically upward at a velocity of 10ms-1 from the top of a tower of
height 50m. After some time the stone passes the edge of the tower and strikes the
ground below the tower. Ignore the effects of friction.
9.1 Draw a labelled free-body diagram showing the force(s) acting on the stone during
its motion. (1)
9.2 Calculate the:
9.2.1 Time taken by the stone to reach its maximum height above the ground (4)
9.2.2 Maximum height that the stone reaches above the ground (4)
9.3 USING THE GROUND AS REFERENCE (zero position), sketch a position time
graph for the entire motion of the stone. (3)
9.4 On its way down, the stone takes 0,1s to pass a window of length 1,5m, as shown in
the diagram above. Calculate the distance (y1) from the top of the window to the
ground. (7)
++++
SUMMATIVE ASSESSMENT (MOMENTUM AND PROJECTILE MOTION)
Study & Master Pages 71/72/73
Page 6 of 6
