HONG KONG CHINESE WOMEN’S CLUB COLLEGE
F.6 APPLIED MATHEMATICS
SUPPLEMENTARY EXERCISE
Selected Examples on Projectile
Motion of a Projectile Under Gravity
1. A particle is projected with a velocity of 30 ms-1 at an elevation of 30o. Find
(a) the greatest height reached,
(b) the time of flight,
(c) the horizontal range,
(d) the velocity and direction of motion at a height of 4 m.
( Ans:
(a) 11.25m
(b) 3s
(c) 77.94m
(d) 28.63ms-1, 24.86o )
2. (a) A rescue plane wants to drop supplies to isolated mountain climbers on a rocky range
ridge 200m below. If the plane is travelling horizontally with a speed of 80ms-1, how far
in advance of the recipients must the goods be dropped?
(b) Suppose, instead, that the plane releases the supplies a horizontal distance of 400m in
advance of the mountain climbers. What vertical velocity should they give the supplies
so that they arrive precisely at the climbers' position? With what speed do the supplies
land in this case? Comment on the results.
( Ans: (a) 506m
(b) -15ms-1, 113.6ms-1 )
3. A particle is projected at an angle . At any time t, the angle of elevation of the particle is ,
and the inclination of the velocity is . Show that
2tan - tan = tan
4. A body is projected so that on its upward path it passes through a point xm horizontal and ym
vertically from the point of projection. Show that, if Rm is the range on a horizontal plane
through the point of projection, the angle of elevation of projection is
 yR 
tan 1 

 x( R  x) 
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5. A ball is projected from a point on the ground distant a from the foot of a vertical wall of
height b, the velocity of projection being V at an angle  to the horizontal.
(a) Find how high above the wall the ball passes it.
(b) If the ball just clear the wall, prove that the greatest height reached is
a 2 tan 2 
4(a tan   b)
( Ans: (a) a tan  
a2 g
b )
2V 2 cos 2 
Determination of Angle of Projection
1. A particle is projected with speed 10ag from a point on level ground at a horizontal
distance 5a from the foot of a vertical pole of height 5a/2, so as just to clear the pole. Find
the possible angles of elevation at the instant of projection.
( Ans: 45o, 71.6o )
2. Show that, if R is the maximum horizontal range for a given velocity of projection, a particle
can be projected to pass through the point whose horizontal and vertical distances from the
point of projection are R/2 and R/4 respectively, provided that the tangent of the angle of
projection is either 1 or 3. Find the range on the horizontal plane in each case.
( Ans: 3R/5, R )
Determination of the Minimum Velocity of Projection
1. Find the least velocity which a projectile may have, so that it may clear a wall, 10m high and
18m from the point of projection. Find also the angle of elevation of the projectile.
( Ans: 17.5ms-1, 59.5o )
2. A projectile is projected under gravity from a point P so as to pass through a point Q whose
horizontal and vertical distances from P are 3 h and h respectively. Prove that the least
speed of projection for which this is possible is 3gh . Prove also that the particle reaches its
highest point before reaching Q, and find its direction of motion at Q.
( Ans: -30o )
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Envelope of Safety
1. A ball is projected so as just to clear two walls, the first of height a at distance b from the
point of projection, and the second of height b at distance a from the point of projection.
Show that the range on a horizontal plane is
a 2  ab  b 2
ab
2. A particle is projected with velocity
2 ga from a point at a height h above a level plain.
Show that the tangent of the angle of elevation for maximum range on the plain is
a
ah
and that the maximum range is 2 a (a+h).
Range on an Inclined Plane
1. (a) A particle is projected with a velocity of 30ms-1 at an angle of 45o to the horizontal. Find
its range on a plane inclined at 30o to the horizontal when projected (i) up, (ii) down the
plane.
(b) For this velocity of projection, find the maximum range (i) up, (ii) down the plane.
( Ans: (a) (i) 43.92m
(b) (i) 60m
(ii) 163.92m
(ii) 180m
)
2. If R is the maximum range on an inclined plane through the point of projection, and T the
corresponding time of flight, show that
R  12 gT 2
3
Inclined Axes
1. A shot is fired with velocity V from a point on an incline  at an elevation + so as to move
in a vertical plane through a line of greatest slope.
(a) Find the range on the incline.
(b) If the shot hits the slope horizontally, show that
tan  
sin  cos
1  sin 2 
2V 2 sin  cos(   )
( Ans:
)
g cos 2 
2. The line OA is inclined at an angle  to the horizontal, with O lower than A and with AO=2a.
A particle is projected from O to pass through A. Its initial velocity v makes an angle  with
OA, and its velocity at A is at right angle to OA. Show that
(a) v2sincos(+) = gacos2,
(b) 2tantan = 1.
If the perpendicular distance of the particle from OA is a when its velocity is parallel to OA,
obtain another relation between tan and tan and deduce that  equals /4 radians.
( Ans:
1
1
 tan   )
tan 
2
3. A particle is projected from a point on an inclined plane, inclined at 2 to the vertical, and
moves towards the upper part of the plane in the vertical plane through a line of greatest slope
of the inclined plane. The initial velocity of the particle is ucos and its initial direction of
motion is inclined at  to the vertical.
(a) Find the time of flight of the particle.
(b) Prove that the range on the inclined plane is u2/2g, the velocity with which it strikes the
plane is usin, and its direction of motion has then turned through a right angle.
( Ans: (a) u/g )
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