SPH3U
Ms. Kueh
Solving Problems with the Big Five Equations
For motion with uniform acceleration, the following formulae can be applied:
The BIG FIVE Kinematics Equations
(1) 𝑣
⃑⃑⃑⃑2 = 𝑣
⃑⃑⃑⃑1 + 𝑎 ∆𝑡
1
(2) ∆𝑑 = 2 (𝑣
⃑⃑⃑⃑1 + ⃑⃑⃑⃑
𝑣2 )∆𝑡
1
(3) ∆𝑑 = 𝑣
⃑⃑⃑⃑1 ∆𝑡 + 2 𝑎∆𝑡 2
1
(4) ∆𝑑 = ⃑⃑⃑⃑
𝑣2 ∆𝑡 − 2 𝑎∆𝑡 2
(5) 𝑣22 = 𝑣12 + 2𝑎∆𝑑
These equations tell us that if we have three pieces of information (such as ⃑⃑⃑⃑
𝑣1 , ⃑⃑⃑⃑
𝑣2 , and ∆𝑡)
regarding an object’s motion, we can determine the other two (𝑎, ∆𝑡).
We can use these equations with horizontal motion or vertical motion.
Example 1
A horse running at 4.0 𝑚/𝑠 accelerates uniformly to a velocity of 18 𝑚/𝑠 in 4.0 𝑠.
What is it’s displacement during the 4.0 𝑠 time interval?
Example 2
Phrancis leaves her house while pushing a stroller south at a constant velocity of
2.7 𝑘𝑚/ℎ but then quickly starts to accelerate 0.40 𝑚/𝑠 2 for 3.6 𝑠. Calculate her
final velocity.
A freely falling object is an object falling under the sole influence of gravity. Freely falling
objects under the influence of gravity accelerate uniformly. On Earth, we can estimate the value
of uniform acceleration, and it is approximately:
𝑔 = 9.8 𝑚/𝑠 2
We let 𝑔 represent the acceleration due to gravity on earth. It’s important to note that in order
to use this value, we must have everything expressed in the appropriate units! For instance, if
you are given a velocity in 𝑘𝑚/ℎ, you must convert this to 𝑚/𝑠 in order to use 𝑔. (Remember to
check this!)
Example 2
A ball is dropped from a balcony 15 𝑚 above the ground. With what velocity
does it strike the ground?
Example 3
The CN Tower in Toronto is 533.33 𝑚 high.
a. How long would it take a nickel, dropped from the top to reach the ground?
b. How fast would the nickel be moving as it hit the ground?
Practice: Equations for motion with Uniform Acceleration
1.
A car acquires a velocity of 32 𝑚/𝑠 by accelerating at 4.0 𝑚/𝑠 2 for 5.0 𝑠. What was its initial
velocity?
2.
A ball falling from rest is located 45 𝑚 below its starting point 3.0 𝑠 later. Assuming that its
acceleration is uniform, what is its value?
3.
The brakes are applied on a car traveling at 30 𝑚/𝑠. The car stops in 3.0 𝑠
a. What is its displacement during this time?
b. What is the car’s average acceleration?
4.
How long will it take a truck traveling at 35 𝑚/𝑠 to stop if it accelerates at – 5.0 𝑚/𝑠 2 ?
5.
A landing plane accelerates at – 15 𝑚/𝑠 2 for 1.0 𝑚 in until it stops. How fast was it going
before it started to slow down?
6.
A car hitting a tree loses 40.0 𝑚/𝑠 in 0.100 𝑠. What is its acceleration?
7.
A skier accelerates at 1.20 𝑚/𝑠 2 down an icy slope, starting from 2.0 𝑚/𝑠. What is her
displacement in 10 𝑠?
8.
What is the acceleration of an object that accelerates steadily from rest, traveling 10 𝑚 in
10 𝑠?
9.
How long does it take an airplane, accelerating from rest at 5.0 𝑚/𝑠 2 to travel a distance of
360 𝑚?
10.
A rocket is moving forward at 120 𝑚/𝑠. When its retro rockets are fired it experiences an
acceleration of – 8.0 𝑚/𝑠 2 . If these rockets are fired for 20 𝑠, determine:
a. The final velocity of the rocket.
b. The displacement of the rocket during the interval.
Answers:
1) 12 𝑚/𝑠
5) 90 𝑚/𝑠
9) 12 𝑠
2) 10 𝑚/𝑠 2
3) 45 𝑚, −10 𝑚/𝑠 2
2
6) – 400 𝑚/𝑠
7) 25 𝑚, 80 𝑚, 165 𝑚
2
10) – 40 𝑚/𝑠 , 800 𝑚
4) 7.0 𝑠
8) 0.20 𝑚/𝑠 2
Physics Problems – Vertical Motion
1. An object is tossed into the air. As it rises, what happens to the acceleration of the
object?
2. A 10 𝑘𝑔 object is dropped from rest.
a. How far will it drop in 2 𝑠?
b. How long will it take a 5 𝑘𝑔 object to drop the same distance?
3. An object is dropped from rest from the top of a 100 𝑚 building. How long will it take
for the object to hit the ground?
4. An object is tossed into the air and allowed to drop to the ground.
a. Sketch a graph of position vs. time for the motion of the object.
b. Sketch a graph of velocity vs. time for the motion of the object.
5. A cliff diver from the top of a 100 𝑚 cliff. He begins his dive by jumping up with a
velocity of 5 𝑚/𝑠.
a. How long does it take for him to hit the water below?
b. What is his velocity right before he hits the water?
6. Alan Iverson slam dunks a basketball and a physics student observes that Iverson’s feet
are 1 𝑚 above the floor at his peak height. At what upward velocity must Iverson leave
the floor to achieve this?
7. A bullet is shot vertically up into the air with a velocity of 450 𝑚/𝑠. Neglecting air
resistance,
a. How long is the bullet in the air?
b. How high does the bullet go?
8. A sandbag is dropped from a hot air balloon that is 300 𝑚 above the ground and rising
at a rate of 13 𝑚/𝑠.
a. How long does it take for the sandbag to hit the ground?
b. How fast is the sand bag going when it hits the ground?
c. At what height is the balloon when the sand bag hits the ground? (Remember
that the balloon is rising at a constant rate while the sand bag is in the air.)
Answers:
2) a. 19.62𝑚 [down]
5) a. 5.06 𝑠
6) 4.4𝑚/𝑠 [up]
8) a. 9.27 𝑠
b. 2 𝑠
3) 4.52 𝑠
b. 44.6𝑚/𝑠 [down]
7) a. 91.8 𝑠
b. 10332 𝑚 [up]
b. 77.8 𝑚/𝑠 [down] c. 420.5 𝑚 [up]