Question

A car is initially at rest on a straight road. The histogram below shows the car's acceleration along that road as a function of time.

Calculate the speed of the car at t = 4 s.

Calculate the distance traveled during the first 5 s.

Calculate the distance traveled from t=10 s to t=13 s.

Calculate the car's average speed from t = 6 s to t=9 s.

Answer 1

1. Calculate the speed of the car at t = 4 s.

Solution: We know that : v = u + a×t

where
V = final velocity
u = initial velocity
a = acceleration
t = time

from t=0 to 1 s a=0, hence v=u + 0*t

as given that u=0 hence v=0

now from  t=1 to 2 s, a= 2 m/s²

v = 0 + 2*(2-1) =2 m/s at t=2 sec.

now from  t=2 to 3 s, a= 3 m/s²

v = 2 + 3*(3-2) = 5 m/s at t=3 sec.

now from  t=3 to 4 s, a= 1 m/s²

v = 5 + 1*(4-3) = 6 m/s at t=4 sec.

hence at t= 4 sec. the speed will be 6 m/sec Ans.

2. Calculate the distance traveled during the first 5 s

Solution: We know that : s = u*t + 0.5*a*t²

where
V = final velocity
u = initial velocity
a = acceleration
t = time
s = distance

We have to calculate distance on a second - by - second basis:

t = 0 s -> 1 s

No distance. The car remains at 0 m/s/s.

t = 1 s -> 2 s

s = ut + 0.5at^2
s = 0.5 (2 m/s/s) (1 s)^2
s = 1 m

v = u + at
v = (2 m/s/s)(1 s)
v = 2 m/s

t = 2 s -> 3 s

s = ut + 0.5at^2
s = (2 m/s)(1 s) + 0.5(3 m/s/s)(1 s)
s = 3.5 m

v = u + at
v = 2 m/s + (3 m/s/s)(1 s)
v = 5 m/s

t = 3 s -> 4 s

s = vt + 0.5at^2
s = (5 m/s)(1 s) + 0.5(1 m/s/s)(1 s)^2
s = 5.5 m

v = u + at
v = 5 m/s + (1 m/s/s)(1 s)
v = 6 m/s

t = 4 s -> 5 s

s = vt + 0.5at^2
s = (6 m/s)(1 s) + 0.5(0 m/s/s)(1 s)^2
s = 6 m

Now add up all the distances:

1 m + 3.5 m + 5.5 m + 6 m = 16 m Ans.

3. Calculate the distance traveled from t=10 s to t=13 s.

from t=5 s to 6 sec

v = u + at

v = 6 +(-1)(1) = 5 m/s

from t=6 s to 8 sec

v = u + at

v = 5 +(1)(2) = 7 m/s

from t=8 s to 10 sec

v = u + at

v = 7 +(-1)(2) = 5 m/s

from t=10 s to 13 sec

v = 5 + 0 =5 m/s

distance s = 5 + 0.5 (0) = 5 m Ans.

4. Calculate the car's average speed from t = 6 s to t=9 s.

speed at t = 6 sec. v= 5 m/s

and speed at t = 9 s v = 5 m/s

hence average speed is 5 m/s