The function ax(t) describes the acceleration of a particle moving along the x-axis. At time t=0, the particle is located at the position x0 and the velocity of the particle is zero.
ax(t)=a0e−bt
The numerical values of all parameters are listed below:
x0=980 m
a0=500 ms2
b=600 s−1
t1=120 s
t2=110 s
Calculate the change in velocity vx between time t=0 and t=t1.
vx= 8.33×10-1 m/s
Calculate the change in position x between time t=0 and t=t1.
x=??
Calculate the average velocity vxave between time t=0 and t=t2.
vxave=???
Calculate the average acceleration of the particle axave between time t=0 and t=t2.
axave= ???
To calculate the change in velocity vx between time t=0 and t=t1, we'll need to integrate the acceleration function ax(t) with respect to time from t=0 to t=t1.
The integral of ax(t) = a0e^(-bt) with respect to t is given by:
∫ ax(t) dt = -(a0/b) * e^(-bt)
Substituting the given values:
a0 = 500 m/s²
b = 600 s⁻¹
t1 = 120 s
we have:
∫ ax(t) dt = -(500/600) * e^(-600t)
Integrating this function from t=0 to t=t1:
vx = ∫[0 to t1] ax(t) dt = -(500/600) * [e^(-600t)] from 0 to 120
To calculate the change in position x between time t=0 and t=t1, we'll need to integrate the velocity function vx(t) obtained from the integration step above.
We integrate the velocity function by taking the integral of vx(t) = ∫ ax(t) dt with respect to t, from t=0 to t=t1.
vx = ∫[0 to t1] ax(t) dt = -(500/600) * [e^(-600t)] from 0 to 120
To calculate the average velocity vxave between time t=0 and t=t2, we need to calculate the change in position x between time t=0 and t=t2, and then divide it by the time elapsed Δt.
vxave = Δx / Δt
To calculate the change in position x between time t=0 and t=t2, we'll need to integrate the velocity function vx(t) with respect to time from t=0 to t=t2.
vxave = Δx / Δt
To calculate the average acceleration axave of the particle between time t=0 and t=t2, we need to calculate the change in velocity vx between time t=0 and t=t2, and divide it by the time elapsed Δt.
axave = Δv / Δt
To calculate Δv, we need to subtract the initial velocity from the final velocity:
Δv = vf - vi
Next, we can calculate Δt by subtracting the initial time from the final time:
Δt = tf - ti
Finally, we can calculate axave:
axave = Δv / Δt