. A ball, initially at rest at t= 0 seconds, rolls with constant acceleration down an inclined plane 10 meters long. If the ball rolls 1 meter in the first 2 seconds, how far will it have rolled at t= 4 seconds?
To find the distance the ball will roll at t=4 seconds, we can use the equations of motion for uniformly accelerated linear motion.
The equation that relates the distance traveled (s), initial velocity (u), time (t), and acceleration (a) is:
s = ut + (1/2)at^2
Given:
Initial velocity, u = 0 m/s
Time, t = 4 s
Acceleration, a = ?
We need to find the acceleration of the ball.
Using the information given, we can determine the acceleration by first finding the average acceleration during the first 2 seconds.
The average acceleration (a_avg) is given by the formula:
a_avg = (change in velocity) / time
We know that the change in velocity (Δv) is equal to the final velocity (v) minus the initial velocity (u).
Since the initial velocity is 0 m/s, the velocity at t=2 seconds is:
v = u + at
Given:
u = 0 m/s
t = 2 s
s = 1 m
We can rearrange the equation to solve for a:
a = (v - u) / t
a = (s - u) / t
Substituting the known values:
a_avg = (1 - 0) / 2
a_avg = 1 / 2
a_avg = 0.5 m/s^2
Since the ball is rolling with a constant acceleration down the inclined plane, the average acceleration during the first 2 seconds is equal to the acceleration throughout the entire motion.
Now, we can use the equation of motion with the known values to find the distance traveled at t=4 seconds:
s = ut + (1/2)at^2
Given:
u = 0 m/s
t = 4 s
a = 0.5 m/s^2
Substituting the known values:
s = (0) + (1/2)(0.5)(4)^2
s = (1/2)(0.5)(16)
s = (1/4)(16)
s = 4 meters
Therefore, the ball will have rolled 4 meters at t=4 seconds.
To solve this problem, we can use the equations of motion for an object with constant acceleration. Let's break it down step by step:
Step 1: Find the acceleration of the ball.
We know that the ball rolls 1 meter in the first 2 seconds. We can use the equation:
d = ut + (1/2)at^2
Where:
d = distance covered (1 meter)
u = initial velocity (0 m/s since the ball is initially at rest)
a = acceleration (unknown)
t = time taken (2 seconds)
Plugging in the values, we get:
1 = 0 + (1/2)a * (2^2)
1 = 2a
Solving for a, we find that the acceleration is 0.5 m/s^2.
Step 2: Find the distance the ball will have rolled at t=4 seconds.
Again, we can use the equation:
d = ut + (1/2)at^2
Where:
d = distance covered (unknown)
u = initial velocity (0 m/s)
a = acceleration (0.5 m/s^2)
t = time taken (4 seconds)
Plugging in the values, we get:
d = 0 + (1/2) * 0.5 * (4^2)
d = 0 + 0.5 * 0.5 * 16
d = 0 + 4
d = 4 meters
Therefore, at t=4 seconds, the ball will have rolled 4 meters.