SME
Grade 9 Math · Problem type
A ball thrown up
Use this for “A ball is thrown up at 19.6 m/s from 2 m. When is it highest, and how high?”
Chart
Vertex (2, 21.6), axis of symmetry t = 2
Zero:
t
=
14 + 6
√
6
7
≈
4.0996
h(1) = 16.7
Landing time: 4.0996 s
This is an example. Type your own values to start a new problem.
g
Gravity
example · m/s²
v
Launch speed
example
h₀
Starting height
example
t
Time
example
h
Height
calculated
a
t² coefficient, −½g
worked out (not typed) · m/s²
tₜₒₚ
Time at the top
worked out (not typed)
hₘₐₓ
Greatest height
worked out (not typed)
tland
Landing time
worked out (not typed)
Formulas
a
=
−
g
2
−4.9
=
−
9.8
2
h
=
a
×
t
2
+
v
×
t
+
h₀
16.7
=
−4.9
×
1
2
+
19.6
×
1
+
2
tₜₒₚ
=
v
g
2
=
19.6
9.8
hₘₐₓ
=
h₀
+
v
2
2 ×
g
21.6
=
2
+
19.6
2
2 × 9.8
tland
=
v
+
√
v
2
+ 2 ×
g
×
h₀
g
4.0996
=
(19.6
+
√
19.6
2
+ 2 × 9.8 × 2
)
÷
9.8
Assumptions
•
On Earth g = 9.8 m/s², so the t² term is −½gt² = −4.9t²; change g for another planet.
•
The top is halfway between the times the ball is at any one height, at t = v ÷ g.
•
The ball lands when h = 0; take the positive time.
Step-by-step
WE KNOW
t = 1 s, g = 9.8 m/s², v = 19.6 m/s, h₀ = 2 m
FIND
t² coefficient, −½g (a), height (h), time at the top (tₜₒₚ), greatest height (hₘₐₓ), landing time (tland)
Step 1 · Find t² coefficient, −½g (a)
Use a = −g ÷ 2
Gravity’s
pull
makes
the
t
2
term
−
1
2
g
.
a
=
−
9.8
2
a
=
−4.9
m/
s
2
Step 2 · Find height (h)
Use h = a × t² + v × t + h₀
Put the time into the height rule.
h
=
−4.9
×
1
2
+
19.6
×
1
+
2
h
=
−4.9
+
19.6
+
2
h
=
16.7
m
Step 3 · Find time at the top (tₜₒₚ)
Use tₜₒₚ = v ÷ g
The
top
is
on
the
axis
of
symmetry,
t
=
−b/(
2
a
)
=
v
g
.
tₜₒₚ
=
19.6
9.8
tₜₒₚ = 2 s
Step 4 · Find greatest height (hₘₐₓ)
Use hₘₐₓ = h₀ + v² ÷ (2 × g)
Put
the
top’s
time
into
the
height
rule;
it
simplifies
to
h₀
+
v
2
2
g
.
hₘₐₓ
=
2
+
19.6
2
2 × 9.8
hₘₐₓ
=
2
+
19.6
2
19.6
hₘₐₓ
=
2
+
384.16
19.6
hₘₐₓ = 2 + 19.6
hₘₐₓ = 21.6 m
Step 5 · Find landing time (tland)
Use tland = (v + √(v² + 2 × g × h₀)) ÷ g
Set
h
=
0
and
use
the
quadratic
formula;
the
positive
root
is
the
landing.
tland
=
(19.6
+
√
19.6
2
+ 2 × 9.8 × 2
)
÷
9.8
tland
=
(19.6
+
√
384.16 + 2 × 9.8 × 2
)
÷
9.8
tland
=
(19.6
+
√
384.16 + 19.6 × 2
)
÷
9.8
tland
=
19.6 +
√
384.16 + 39.2
9.8
tland
=
19.6 +
√
423.36
9.8
tland
=
19.6 + 20.58
9.8
tland
=
40.18
9.8
tland = 4.0996 s
Check
Put the numbers back in. Both sides should match.
−4.9
=
−
9.8
2
✓
16.7
=
−4.9
×
1
2
+
19.6
×
1
+
2
✓
2
=
19.6
9.8
✓
21.6
=
2
+
19.6
2
2 × 9.8
✓
4.0996
≈
(19.6
+
√
19.6
2
+ 2 × 9.8 × 2
)
÷
9.8
✓
Topic check
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