Acceleration
𝑣=
∆𝑥
∆𝑡
Name__________________
; 𝑎 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 =
∆𝑣
∆𝑡
1
; ∆𝑥 = 𝑣𝑜 ∆𝑡 + 2 𝑎(∆𝑡)2
1. What is the acceleration of Andrei’s purple Toyota if its speed is increased
uniformly from 44 m/s to 66 m/s over an 11 second period?
𝑣 −𝑣𝑜
𝑚
𝑚
∆𝑣
𝑚
𝑣𝑓 = 66 𝑠 ; 𝑣𝑜 = 44 𝑠 ; ∆𝑡 = 11 sec ; 𝑎 = ∆𝑡 = 𝑓∆𝑡 = 2 𝑠2
2. What is the acceleration of Jenna’s puppy Fluffy if its speed is decreased
uniformly from 76 m/s to 34 m/s in 7 seconds? ( Fluffy is one speedy dog!)
𝑣 −𝑣𝑜
𝑚
𝑚
∆𝑣
𝑚
𝑣𝑓 = 34 𝑠 ; 𝑣𝑜 = 76 𝑠 ; ∆𝑡 = 7 sec ; 𝑎 = ∆𝑡 = 𝑓∆𝑡 = −6 𝑠2
3. A car is halted at a stop light. When the light turns green the driver accelerates at
the rate of 9 m/s2 for 7 seconds. How fast will the car be traveling at the end of the
seven seconds?
𝑚
𝑚
∆𝑣
𝑣𝑜 = 0 𝑠 ; ∆𝑡 = 7 sec ; 𝑎 = 9 𝑠2 ; 𝑎 = ∆𝑡 → ∆𝑣 = 𝑎𝑡 → 𝑣𝑓 − 𝑣𝑜 = 𝑎∆𝑡 → 𝑣𝑓 =
𝑚
𝑣𝑜 + 𝑎∆𝑡 = 63 𝑠
4. A car accelerated at a rate of 3 ft/s2 for 12 seconds. At the end of this time period
the velocity of the car was 55 ft/s. What was the car’s velocity before it started to
accelerate?
𝑓𝑡
𝑓𝑡
∆𝑣
𝑣𝑓 = 55 𝑠 ; ∆𝑡 = 12 sec ; 𝑎 = 3 𝑠2 ; 𝑎 = ∆𝑡 → ∆𝑣 = 𝑎𝑡 → 𝑣𝑓 − 𝑣𝑜 = 𝑎∆𝑡 →
𝑓𝑡
𝑣𝑜 = 𝑣𝑓 − 𝑎∆𝑡 = 19 𝑠
5. How far will a chicken on roller skates travel if she starts at a velocity of 5 m/s
and ends at a velocity of 25 m/s, over a time frame of 7.5 seconds?
𝑣 −𝑣𝑜
𝑚
𝑚
∆𝑣
𝑚
𝑣𝑓 = 25 𝑠 ; 𝑣𝑜 = 5 𝑠 ; ∆𝑡 = 7.5 sec ; 𝑎 = ∆𝑡 = 𝑓∆𝑡 = 2.67 𝑠2
1
∆𝑥 = 𝑣𝑜 ∆𝑡 + 2 𝑎(∆𝑡)2 = 112.5 𝑚
6. A zebra needs to cover 100 m in 7 seconds to escape a lion. If he starts from rest,
what final velocity does he need to attain at the end of 7 seconds?
𝑚
1
∆𝑣
∆𝑥 = 100 𝑚 ; 𝑣𝑜 = 0 𝑠 ; ∆𝑡 = 7 sec ; ∆𝑥 = 𝑣𝑜 ∆𝑡 + 2 𝑎(∆𝑡)2 ; 𝑎 = ∆𝑡 → 𝑎∆𝑡 =
1
1
2∆𝑥
2
2
∆𝑡
∆𝑣 = 𝑣𝑓 − 𝑣𝑜 → ∆𝑥 = 𝑣𝑜 ∆𝑡 + (𝑣𝑓 − 𝑣𝑜 )∆𝑡 = (𝑣𝑓 + 𝑣𝑜 )∆𝑡 → 𝑣𝑓 =
𝑣𝑜 = 28.6
𝑚
−
𝑠
7. If two sprinters accelerate at the same rate, but one runs for 4 times as long, how
much faster will they be running compared to the other runner?
∆𝑣
𝑎 = ∆𝑡 → ∆𝑣 = 𝑎∆𝑡 ; if one sprinter runs 4 times longer, or ∆𝑡1 = 4∆𝑡, then they
will be moving at ∆𝑣1 = 𝑎∆𝑡1 = 𝑎(4∆𝑡) = 4∆𝑣, or 4 times faster
8. Two identical cars are driving along the road. One is moving twice as fast as the
other. If both slam on the breaks at the same time, how much longer will it take
the faster moving car to completely stop?
Similar reasoning as in problem 7 above but if the change in velocity is twice as
much, then the time to stop is twice as long.
Velocity vs. Time
8
7
6
v (m/s)
5
X
4
Y
3
2
1
0
0
2
4
6
8
10
time (seconds)
.
12. In the graph above, velocity is graphed versus time.
a. What is the acceleration of vehicle X?
Acceleration is the slope of a velocity vs. time graph. The slope of vehicle x’s graph
𝑚
is zero so 𝑎𝑥 = 0 𝑠2
b. What is the acceleration of vehicle Y?
3𝑚
3𝑚
Same reasoning as above in part a. The slope of vehicle y’s graph is 4 𝑠2 so 𝑎𝑦 = 4 𝑠2
c. At the time when the vehicles have the same velocity, find the distance X has traveled.
𝑚
1
𝑣0 = 6 𝑠 ; They are at the same velocity at ∆𝑡 = 8 sec ; ∆𝑥 = 𝑣𝑜 ∆𝑡 + 2 𝑎(∆𝑡)2 and
using the acceleration from part a ∆𝑥 = 48 𝑚
d. What distance will Y have traveled when its velocity is equal to X’s velocity?
𝑚
1
𝑣0 = 0 𝑠 ; They are at the same velocity at ∆𝑡 = 8 sec ; ∆𝑥 = 𝑣𝑜 ∆𝑡 + 2 𝑎(∆𝑡)2 and
using the acceleration from part a ∆𝑥 = 24 𝑚
13. Sketch position vs. time for both of the vehicles in problem 12 above on the same
graph.
See next page
x
Vehicle x, slope should be 6
Vehicle y, slope should be
increasing
t