An arrow is accelerated for a displacement of 75cm [fwd] while it is on the bow. If the arrow leaves the bow at a velocity of 75m/s [fwd], what is the average acceleration while on the bow?
Displacement = 75cm = 0.75m
Initial velocity = 0
Velocity 2 = 75 m/s [fwd]
Acceleration while on the bow = ?
Use the following equation to solve for time.
d = ( (v2 + v1)/2 ) x t
t = d(2/(v2+v1))
t = 0.75(2/(75+0))
t = 0.02s
Plug all the values we have in the following kinematics equation for acceleration.
v2 = v1 + at
a = (v2 - v1)/t
a = (75 - 0)/0.02
a = 3750 m/s^2
Therefore, the acceleration is 3.8 x 10^3 m/s^2 [fwd]
Well, if the arrow is accelerated over a displacement of 75 cm and it ends up with a velocity of 75 m/s, it means it really wanted to get away from that bow in a hurry! The average acceleration can be calculated using the equations of motion.
The equation for average acceleration is:
acceleration = (final velocity - initial velocity) / time
Since we only have the final velocity and the displacement, we'll need to find the time it took for the arrow to travel that distance. Thankfully, there's another equation that can help us:
displacement = (initial velocity * time) + (0.5 * acceleration * time^2)
Hmm, it looks like we need the initial velocity to calculate the acceleration, and we need the acceleration to calculate the initial velocity!
Well, we might need to go undercover and interrogate the arrow to find its initial velocity, or maybe it has a secret diary hidden in its quiver that will give us some clues. Or, you know, you could have accidentally left out some information in the question... It happens to the best of us!
To calculate the average acceleration, we need to use the formula:
average acceleration = change in velocity / change in time
In this case, we are given the displacement of the arrow on the bow, but we are not given the time it takes for the arrow to reach the velocity of 75m/s. However, we can calculate the time using the displacement and velocity.
To find the time, we can use the following kinematic equation:
final velocity^2 = initial velocity^2 + 2 * acceleration * displacement
Since we know the initial velocity (0 m/s), final velocity (75 m/s), and the displacement (75 cm = 0.75 m), we can rearrange the equation to solve for acceleration:
acceleration = (final velocity^2 - initial velocity^2) / (2 * displacement)
Substituting the values:
acceleration = (75 m/s)^2 / (2 * 0.75 m)
Simplifying:
acceleration = (5625 m^2/s^2) / (1.5 m)
acceleration = 3750 m/s^2
Therefore, the average acceleration of the arrow while on the bow is 3750 m/s^2.
a = change in speed / time
average speed = (75/2 )meters/ second
so time = distance/speed = .75meters/(75/2 meters/second) =2*10^-4 seconds
acceleration = 75/2*10^-4 = 372500 m/s^2