Question : If a is dependent on v as follows :
a=v and at t=0, x=1m, v=1m/s
a) Find out its velocity at t=2 s
b) Find out its velocity at x=3m.
Doubt by Vanshika
Solution :
a=v (Given)
Initially at t=0
x = 1 m & v=1m/s
a) Please note that here the objective is to find the expression of velocity in terms of time.
a=v (Given)
dv/dt = v
dv = vdt
dv = vdt
dv/v = dt
Integrating both sides within proper limits
Integrating both sides within proper limits
1∫vdv/v = 0∫2dt
ln|v|1v = [t]02
ln|v| - ln |1| = [2-0]
ln|v| - 0 = 2 [∵ ln|1|=0]
ln|v| = 2
v = e2
v = e2 m/s
Hence, the velocity at t=2 sec is e2 m/s.
ln|v| - ln |1| = [2-0]
ln|v| - 0 = 2 [∵ ln|1|=0]
ln|v| = 2
v = e2
v = e2 m/s
Hence, the velocity at t=2 sec is e2 m/s.
b) Please note that here the objective is to find the expression of velocity in terms of displacement.
a=v (Given)
dv/dt = v
(dv/dt) × (dx/dx) = v
(dv/dx) × (dx/dt) = v
(dv/dx) × v = v [∵dx/dt = v]
dv/dx = v/v
dv/dx = 1
(dv/dt) × (dx/dx) = v
(dv/dx) × (dx/dt) = v
(dv/dx) × v = v [∵dx/dt = v]
dv/dx = v/v
dv/dx = 1
dv = dx
Integrating both sides
1∫vdv = 1∫3dx
[v]1v = [x]13
[v-1] = [3-1]
v-1 = 2
v = 2+1
v = 3 m/s
Integrating both sides
1∫vdv = 1∫3dx
[v]1v = [x]13
[v-1] = [3-1]
v-1 = 2
v = 2+1
v = 3 m/s
Hence, the velocity at x=3 m is 3 m/s.