The power needed to accelerate a projectile from rest to its launch speed v in a time t is 48.0 W. How much power is needed to accelerate the same projectile from rest to a launch speed of 2v in a time of 1/2t
To find the power needed to accelerate the projectile from rest to 2v in a time of 1/2t, we can use the formula for power:
Power = Work / Time
First, let's determine the work done in accelerating the projectile from rest to 2v. The work done is equal to the change in kinetic energy:
Work = ΔKE = KE_f - KE_i
Since the projectile starts from rest, the initial kinetic energy (KE_i) is zero. The final kinetic energy (KE_f) can be calculated using the formula:
KE = (1/2)mv^2
Since the launch speed is 2v, the final kinetic energy can be written as:
KE_f = (1/2)m(2v)^2 = 2mv^2
Therefore, the work done in accelerating the projectile from rest to 2v is:
Work = 2mv^2
Next, let's calculate the time required to accelerate the projectile from rest to 2v in a time of 1/2t. Given that the original time required to accelerate the projectile from rest to v is t, the time required to accelerate it to 2v in 1/2t would be half of t, which is t/2.
Finally, we can substitute the values into the power formula:
Power = Work / Time
Power = (2mv^2) / (t/2)
Power = 4mv^2 / t
Now, let's substitute the values from the given information. The power needed to accelerate the projectile from rest to v in time t is 48.0 W. We can write this as:
48.0 W = 4mv^2 / t
To find the power needed to accelerate the same projectile to 2v in 1/2t, we can rearrange the equation:
Power = 4mv^2 / t = (48.0 W) * (2t / t) = 96.0 W
Therefore, the power needed to accelerate the same projectile from rest to a launch speed of 2v in a time of 1/2t is 96.0 W.