Two objects moving with constant acceleration starting from rest are to cover the same distance. If the second is to cover that distance in ¼ of the time would need
a: a quarter of the first’s acceleration.
b: a half of the first’s acceleration.
c: sixteen times the first’s acceleration.
d: eight times the first’s acceleration.
distance=1/2 a t^2
a= 2 distance/t^2
so if the new t is 1/4, the new a must be 16 times
To determine how the acceleration of the second object compares to the first, we can use the kinematic equation for displacement:
d = (1/2) * a * t^2
where d is the distance covered, a is the acceleration, and t is the time.
Let's assume the distance covered is represented by "d" for both objects. The first object covers the distance d in time t, while the second object covers the same distance in 1/4 of that time, which is represented as (1/4)t.
Using the kinematic equation, we can set up the following equation for the first object:
d = (1/2) * a * t^2
And for the second object:
d = (1/2) * a2 * (1/4t)^2
Simplifying the second equation:
d = (1/2) * a2 * (1/16) * t^2
Comparing the two equations, we can see that the distance covered is the same (d = d) and the time squared terms cancel out. Thus, we have:
(1/2) * a * t^2 = (1/2) * a2 * (1/2^4) * t^2
Simplifying further:
a = a2/2^4
This shows that the acceleration of the second object (a2) is 1/2^4 (or 1/16) times the acceleration of the first object (a). Therefore, the answer is c: sixteen times the first's acceleration.